Practice Test #2 Data Sufficiency (218 Questions) 1. Is m > k ? (1) 3m > 3k (2) 2m > 2k (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: m>k? (1) 即m>k時，所以可以判定，sufficient。 (2) 即m>k時，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------2. If k is a positive integer, is k the square of an integer? (1) k is divisible by 4. (2) k is divisible by exactly four different prime numbers. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: k是整數平方? (1)即k=4a=22 * a時，不知道a，所以無法判定，insufficient。 (2)即k有4個質因數時，k=2x * 3y * 5z *7w，所以無法判定，insufficient。 (3)由(1)(2)合併，只知x至少有兩個2，還是無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------3. If w and z are positive, is w/z < 1? (1) w < z (2) z < 4 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】A 【思路】題目問: w/z < 1? (1) 即w<z時，又w, z為正，所以可以判定，sufficient。 (2) 即z<4時，不知道w，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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4. The sale price of a certain jacket was 15 percent less than its original price, and the sale price of a certain shirt was 10 percent less than its original price. How much greater was the original price of the jacket than the original price of the shirt? (1) The sale price of the jacket was \$83 greater than the sale price of the shirt. (2)

The original price of the jacket was \$140.

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: j - s=? Jacket: original price為j；sale price為0.85j； Shirt: original price為s；sale price為0.90s； (1) 即0.85j - 0.9s=83時，所以無法判定，insufficient。 (2) 即j=140時，不知道s值，所以無法判定，insufficient。 (3) 由(1)(2)合併，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------5. If x, y, and z are integers, is x + y + 2z even? (1) x + z is even. (2) y + z is even. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: x + y + 2z為偶數? (1) 即x + z為偶時，不知道y，所以無法判定，insufficient。 (2) 即y + z為偶時，不知道x，所以無法判定，insufficient。 (3)由(1)(2)合併時，兩者相加即為所求，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------6. The operation @ represents either addition, subtraction, or multiplication of integers. What is the value of 1 @ 0 ? (1) 0 @ 2 = 2 (2) 2 @ 0 = 2 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: 1 @ 0=? (1) 即0 @ 2 = 2時，@只有為+才有可能，代回1 @ 0，得知所求為1，所以可以判定，sufficient。 (2) 即2 @ 0 = 2時，@為+, -都有可能，代回1 @ 0，得知所求都為1，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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7. One kilogram of a certain coffee blend consists of x kilogram of type I coffee and y kilogram of type II coffee. The cost of the blend is C dollars per kilogram, where C = 6.5x + 8.5y. Is x < 0.8? (1) y > 0.15 (2) C ≧ 7.30 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B 【思路】題目問: x < 0.8? 已知 x + y = 1 (1) 即y>0.15時，x<0.85，所以無法判定，insufficient。 (2) 即6.5x+8.5y≧7.3時，y=1-x代入，得到x≦0.6，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------8. If n and p are integers, is p > 0? (1) n + 1 > 0 (2) np > 0 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: p>0? (1) 即n+1>0時，不知道p，所以無法判定，insufficient。 (2) 即np>0時，不知道n的正負，所以無法判定，insufficient。 (3) 由(1)(2)合併，n>-1，又n為整數，np>0 => n≧1，=> p>0，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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9. Each of the 105 students in a certain club is either a freshman, a sophomore, or a junior. club are sophomores?

How many of the students in the

(1) The ratio of the number of freshmen to the number of sophomores is 1 to 2. (2) The ratio of the number of freshmen to the number of juniors is 1 to 4. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: sophomore人數? 已知 f+s+j = 105，問s=? (1) 即f:s=1:2時，不知道junior關係，所以無法判定，insufficient。 (2) 即f:j=1:4時，不知道sophomore關係，所以無法判定，insufficient。 (3)由(1)(2)合併，f:s:j=1:2:4，=> s=105 * [2/(1+2+4)] = 30，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------10. Juan bought some paperback books that cost \$8 each and some hardcover books that cost \$25 each. If Juan bought more than 10 paperback books, how many hardcover books did he buy? (1) The total cost of the hardcover books that Juan bought was at least \$150. (2) The total cost of all the books that Juan bought was less than \$260. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: 買了多少本hardcover? 已知8p+25h，且p>10，問h=? (1) 即25h≧150時，h≧6，所以無法判定，insufficient。 (2) 即8p+25h<260時，所以無法判定，insufficient。 (3) 由(1)(2)合併，h≧6，又p>10 => p=11時有最小值，此時h有最大值 => h<6.88，=> h=6，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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11. 錯 What is the value of the positive integer m? (1) When m is divided by 6, the remainder is 3. (2) When 15 is divided by m, the remainder is 6. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B 【思路】題目問: m=? (1) 即m=6a+3時 m可能為3, 9, 15 所以無法判定，insufficient。 (2) 即15=bm+6時 注意，餘數會小於除數---非常重要! bm=9，且m>6，=> m=9 所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------12. Sets A, B, and C have some elements in common. If 16 elements are in both A and B, 17 elements are in both A and C, and 18 elements are in both B and C, how many elements do all three of the sets A, B, and C have in common? (1) Of the 16 elements that are in both A and B, 9 elements are also in C. (2) A has 25 elements, B has 30 elements, and C has 35 elements. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】A 【思路】題目問: A∩B∩C=? A∩B=16，A∩C=17，B∩C=18 (A∪B∪C) = A + B + C - (A∩B) – (B∩C) – (C∩A) + (A∩B∩C) (1) 即A∩B∩C=9時，所以可以判定，sufficient。 (2) 即A=25，B=30，C=35時，不知道(A∪B∪C)，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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13. Of the people who attended a workshop, 60 percent were teachers and some of the teachers were teachers of language arts. What percent of the people who attended the workshop were teachers of language arts? (1) 200 people attended the workshop. (2) 72 of the teachers who attended the workshop were not teachers of language arts. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: 語言老師在研討會所佔的比例? (1) 即200人參加研討會時 可知200*60%=120為老師 但不知道語言老師佔所有老師的比例 所以無法判定，insufficient。 (2) 即72位參加研討會老師不是語言老師時 但不知道語言老師人數和總人數 所以無法判定，insufficient。 (3) 由(1)(2)合併，語言老師=120-72=48 48/200=24%，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------14. Jane walked for 4 miles. What was her average speed for the first 2 miles? (1) Jane's average speed for the 4 miles was 3.2 miles per hour. (2) It took Jane 15 minutes longer to walk the second 2 miles than it took her to walk the first 2 miles. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: 前2 miles平均速度? (1) 即全程4miles平均速度是3.2m/h時 全程時間=4/3.2=1.25(hr)，但不知道前2 miles時間，所以無法判定，insufficient。 (2) 即後2miles比前2miles多花15分鐘時 不知道前2 miles時間，所以無法判定，insufficient。 (3) 由(1)(2)合併，1.25(hr)=75(min)， 又x+(x+15)=75，=> x=30，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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15. Is xy + xz = 0? (1) x = 0 (2) y + z = 0 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: xy + xz = 0? 已知xy + xz =x(y+z) (1) 即x=0時，所以可以判定，sufficient。 (2) 即y+z=0時，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------16. Is |k | = 2? (1) k2 = 4 (2) k = |-2| (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: |k|=2? (1) 即k=+2或-2時，所以可以判定，sufficient。 (2) 即k=|-2|時，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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17. 錯 Store S sold a total of 90 copies of a certain book during the seven days of last week, and it sold different numbers of copies on any two of the days. If for the seven days Store S sold the greatest number of copies on Saturday and the second greatest number of copies on Friday, did Store S sell more than 11 copies on Friday? (1) Last week Store S sold 8 copies of the book on Thursday. (2) Last week Store S sold 38 copies of the book on Saturday. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B 【思路】題目問: 第二大值會大於11? x1+x2+x3+x4+x5+x6+x7=90，其中x6有最大值，x5有第二大值，任兩數不相同，問x5>11? 直接代入數字檢驗最快 (1) 即x4=8時 x1+x2+x3+x5+x6+x7=82 x5有最小值時，討論除了x6外，其他4個數x1, x2, x3, x7越平均時的情形即可 (i) (0, 1, 2, 3)時，又x4=8，且x5為第二大值，所以x5=9 (ii) (9, 10, 11, 12)時，又x4=8，且x5為第二大值，所以x5=13 所以無法判定，insufficient。 (2) 即x6=38時 x1+x2+x3+x4+x5+x7=52 x5有最小值時，其他5個數x1, x2, x3, x4, x7越平均且有最大值時的情形即可 52/6=8.xxx (i) (6, 7, 8, 9, 10)時，又x5為第二大值 所以x5=12可以成立 (ii) (7, 8, 9, 10, 11)時，又x5為第二大值 所以x5=7無法成立 (iii)如果x5=11為最小值時 (10, 9, 8, 7, 6) => total是51(不合，總和是52) 所以x5=11，不能成立 (iv)如果x5=10為最小值時 (9, 8, 7, 6, 5) => total是45(不合，總和是52) 所以x5=10，不能成立 以此類推，所以x5最小值=12，可以判定，sufficient。 (代數字進去檢驗，會更有感覺)。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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18. Are the two nonzero integers x and y on opposite sides of 0 on the number line? (1) The sum of x and y is 0. (2) The product of x and y is less than 0. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: x, y在數線的不同邊? 已知x, y為兩非零整數 (1) 即x+y=0時，必為一正一負，所以可以判定，sufficient。 (2) 即xy<0時，必為一正一負，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------19. 錯 Set S consists of five consecutive integers, and set T consists of seven consecutive integers. Is the median of the numbers in set S equal to the median of the numbers in set T? (1) The median of the numbers in set S is 0. (2) The sum of the numbers in set S is equal to the sum of the numbers in set T. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: 集合S和集合T的中位數相等? 設集合S有5個連續整數，m-2, m-1, m, m+1, m+2 集合T有7個連續整數，n-3, n-2, n-1, n, n+1, n+2, n+3 (1) 即集合S的中位數=0時 即m=0，不知道n 所以無法判定，insufficient。 (2) 即集合S的總和=集合T的總和時 即5m=7n，(m, n)可以為(7,5)或(0,0) 所以無法判定，insufficient。 (3)由(1)(2)合併，得到n=0，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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20. 錯 If -2x > 3y, is x negative? (1) y > 0 (2) 2x + 5y - 20 = 0 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問:若-2x>3y，則x<0? (1) 即y>0時 又-2x>3y，則x<0 所以可以判定，sufficient。 (2) 即2x+5y=20時 則2x+3y+2y=20 已知 2x+3y < 0，所以y>0 再代回-2x > 3y，可得出x<0 所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------21. 錯 Is the integer n odd? (1) n is divisible by 3. (2) 2n is divisible by twice as many positive integers as n. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B 【思路】題目問:n是奇數? 直接代入數字檢驗最快 (1) 即n可以被3整除時 n可以為3, 6, 9 所以無法判定，insufficient。 (2) 即可以整除2n的正整數是n的兩倍時 (i) n=9，則有1, 3, 9 => 3個正整數 2n=18，則有1, 2, 3, 6, 9, 18 => 6個正整數 (ii)n=8，則有1, 2, 4, 8 => 4個正整數 2n=16，則有1, 2, 4, 8, 16 => 5個正整數 其中新增的數字2=1*2，6=3*2，18=9*2，分別都是可被n整除數字的2倍。 因此可以推論出: n其中必定不能被2整除 所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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22. 錯 Is 1/p > r / (r2 +2) (1) p = r (2) r > 0 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: 1/p > r/(r2 +2)? (1) 即p=r時 若1/r > r/(r2 +2) => 1/r - r/(r2 +2) >0 2/ r*( r2 +2) > 0 (i) r>0時，可以成立 (ii) r<0時，無法成立 所以無法判定，insufficient。 (2) 即r>0時 不知道p值正負 所以無法判定，insufficient。 (3) 由(1)(2)合併或1/r > r/(r2 +2) => r/r2 > r/(r2 +2) 所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------23.

q q q q

q r u x

q s v y

q t w z

In the table above, is z = 20q? (1) q = 3 (2) Each value in the table other than q is equal to the sum of the value immediately above it in the table and the value immediately to its left in the table. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B 【思路】題目問: z=20q? (1) 即q=3時 不知道z值，所以無法判定，insufficient。 (2)每個值是其上面的數字和左邊的數字的總和， z=w+y=(t+v)+(x+v)=x+2v+t =(u+q)+2(u+s) +(s+q)= 2q+3u+3s =2q+3(q+r)+3(q+r)=8q+6r =8q+6(2q) =20q 所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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24. One member of a committee of 5 men and 8 women resigned and was not replaced. What fraction of the remaining members were men? (1) 7/12 of the remaining members were women. (2) The member who resigned was a woman. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: 剩下男生比例? (1) 即1 - (7/12) = 5/12，所以可以判定，sufficient。 (2) 即5/(13-1) = 5/12，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------25. If x and y are greater than 0, is x = 1? (1) x/y = 1 (2) xy = 1 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: x=1? 已知x, y >0，問x=1? (1) 即x/y =1時，可知x=y，所以無法判定，insufficient。 (2) 即xy =1時，x=1/y，不知道y，所以無法判定，insufficient。 (3) 由(1)(2)合併，=> x2=1，=> 因為x>0，x=1，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------26. If x and y are integers, is the value of x(y + 1) even? (1) x and y are prime numbers. (2) y > 7 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: x(y+1)為偶數? (1) 即x, y為質數時 考慮x為質數2的情形，所以無法判定，insufficient。 (2) 即y>7時，不知道x, y的值，所以無法判定，insufficient。 (3)由(1)(2)合併時，y+1必為偶數，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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27. Does set S contain any even numbers? (1) There are no prime numbers in S. (2) There are no multiples of 4 in S. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: 集合S包含偶數? 直接代數字下去判斷最快 (1) 即集合S不包含質數時 S={4, 9} or {9, 15}，所以無法判定，insufficient。 (2) 即集合S沒有4的倍數時 S={6, 9} or {9, 15}，所以無法判定，insufficient。 (3)由(1)(2)合併，S={6, 9} or {9, 15}，所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------28. A certain movie depicted product A in 21 scenes, product B in 7 scenes, product C in 4 scenes, and product D in 3 scenes. The four product manufacturers paid amounts proportional to the number of scenes in which their product was depicted in the movie. If each manufacturer paid x dollars per scene, how much did the manufacturer of product D pay for this advertising? (1) The manufacturers of products A and B together paid a total of \$560,000 for this advertising. (2) The manufacturer of product B paid \$60,000 more for this advertising than the manufacturer of product C paid. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: D付多少廣告費? 已知 A:B:C:D = 21:7:4:3 (1) 即A+B=560,000時 D=560,000*[3/(21+7)]=60,000，所���可以判定，sufficient。 (2) 即B-C=60,000時 D=60,000*[3/(7-4)]=60,000，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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29. Is the positive integer n odd? (1) n = 2k + 1, where k is a positive integer. (2) 2n + 1 is an odd integer. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】A 【思路】題目問: 正整數n為奇數? (1)即n=2k+1時，恆成立，所以可以判定，sufficient。 (2)即2n+1為奇數時，n可以為奇或偶，所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------30. If n is an integer, is n/7 an integer? (1) 3n/7 is an integer. (2) 5n/7 is an integer. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: n/7是整數? (1) 即3n/7是整數時 n是7的倍數，所以可以判定，sufficient。 (2) 即5n/7是整數時 n是7的倍數，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------31. If a wire 27 meters long is cut into three pieces of three different lengths, what is the length of the longest piece? (1) The length of the longest piece is twice the length of the shortest piece. (2) The sum of the lengths of the two shorter pieces is 15 meters. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B 【思路】題目問: 切成三段不同長度後的最大值? 已知a+b+c=27，假設a < b< c，問c=? (1)即c=2a時，不知道b，所以無法判定，insufficient。 (2)即a+b=15時，=> c=12，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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32. Is positive integer n divisible by 3? (1) n2 / 36 is an integer. (2) 144 / n2 is an integer. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】A 【思路】題目問: n可以被3整除? (1) 即n2除以36可以整除時 又36=22 * 32，=> n一定包含2和3，所以可以判定，sufficient。 (2) 即144除以n2可以整除時 又144=24 * 32，=> n不一定包含3，無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------33. 錯 If a > 0, b > 0, and c > 0, is a(b - c) = 0 ? (1) b – c = c - b (2) b/c = c/b (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: a(b - c) = 0? (1) 即b=c時 所以可以判定，sufficient。 (2) 即b=+c或-c時 又b>0，c>0，所以b=+c 所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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34. Are positive integers p and q both greater than n? (1) p - q is greater than n . (2) q > p (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: 正整數p和q都大於n? (1) 即p - q> n時 只知p>n，但不知道q和n的關係，所以無法判定，insufficient。 (2) 即q>p時 不知道和n的關係，所以無法判定，insufficient。 (3)由(1)(2)合併，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------35. What is the value of xy? (1) y = x + 1 (2) y = x2 + 1 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: xy的值? (1) 即y=x+1時 不知道x值，所以無法判定，insufficient。 (2) 即y=x2 +1時， 不知道x值，所以無法判定，insufficient。 (3) 由(1)(2)合併，x+1=x2+1，=> x(x-1)=0 => x=0或1 => (x,y)=(0, 1)或(1, 2) => xy=0或2，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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36. If ax + b = 0, is x > 0? (1) a + b > 0 (2) a - b > 0 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: x=-(b/a)>0? (1) 即a+b>0時 (a, b)可能為(+,+), (+,-), (-,+) 所以無法判定，insufficient。 (2) 即a-b>0時 (a, b)可能為(+,+), (+,-), (-,-) 所以無法判定，insufficient。 (3)由(1)(2)合併，(a, b)可能為(+,+),(+,-)，還是無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------37. Are the integers z and f to the right of 0 on the number line? (1) The product of z and f is positive. (2) The sum of z and f is positive. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: 整數z和f在0的右側? 問z>0, f>0? (1) 即zf>0時，(z, f)可能為(+,+)或(-,-)，所以無法判定，insufficient。 (2) 即z+f>0時，(z, f)可能為(+,+),(+,-)或(-,+)，所以無法判定，insufficient。 (3) 由(1)(2)合併，(z, f)只能為(+,+)，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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38. Some of the students enrolled at College T are part-time students and the rest are full-time students. By what percent did the number of full-time students enrolled at College T increase from the fall of 1999 to the fall of 2000? (1) There were 50 more full-time students enrolled at College T in the fall of 2000 than in the fall of 1999. (2) The total number of students enrolled at College T increased by 5 percent from the fall of 1999 to the fall of 2000. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: Full-time students增加的比例? 即F(2000)-F(1999)/ F(1999)=? (1)即F(2000) - F(1999)=50時 不知道F(1999)，所以無法判定，insufficient。 (2)即F(2000)+P(2000)=1.05[F(1999)+ P(1999)]時， 所以無法判定，insufficient。 (3)由(1)(2)合併時，不知道F(1999)，所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------39. If c and d are integers, is c even? (1) c(d + 1) is even. (2) (c + 2)(d + 4) is even. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: c為偶數? (1) 即c(d+1)為偶時 可能c為偶數或(d+1)為偶數 => c為偶數或d為奇數 所以無法判定，insufficient。 (2) 即(c+2)(d+4)為偶時 可能(c+2)為偶數或(d+4)為偶數 => c為偶數或d為偶數 所以無法判定，insufficient。 (3) 由(1)(2)合併，只能c為偶數，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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40. If set S consists of the numbers 1, 5, -2, 8, and n, is 0 < n < 7 ? (1) The median of the numbers in S is less than 5. (2) The median of the numbers in S is greater than 1. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: 0<n<7 ? 由小到大排列為-2, 1, 5, 8，再考慮n的位置 (1)即中位數小於5時，只知n<5，所以無法判定，insufficient。 (2)即中位數大於1時，只知n>1，所以無法判定，insufficient。 (3)由(1)(2)合併，1<n<5，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------41. Last Thursday, John assembled chairs at a rate of 3 chairs per hour for part of the day and Larry assembled no chairs. Last Friday, Larry assembled chairs at a rate of 4 chairs per hour for part of the day and John assembled no chairs. If John and Larry assembled chairs for a total of 7 hours during these two days, how many chairs did John assemble last Thursday? (1) During these two days, John and Larry assembled a total of 25 chairs. (2) During these two days, Larry assembled more chairs than John did. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】A 【思路】題目問: John組了多少張椅子? 已知TJ+TL=7，問3TJ=? (1) 即3TJ+4TL =25時 => TJ =3，所以可以判定，sufficient。 (2) 即4TL>3TJ時，所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------42. What is the value of 3n - 4? (1) 6n – 10 = 30 (2) n/3 = 20/9 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: 3n-4的值? (1) 即6n=40時，=>3n=20時，所以可以判定，sufficient。 (2) 即9n=60時，=>3n=20時，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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43. Circle C and line k lie in the xy-plane. If circle C is centered at the origin and has radius 1, does line k intersect circle C? (1) The x-intercept of line k is greater than 1. (2) The slope of line k is -(1/10). (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: 直線k會與圓C相交? (1) 即直線k的x截距大於1時，所���無法判定，insufficient。 (2) 即直線k的斜率是-(1/10)時，所以無法判定，insufficient。 (3) 由(1)(2)合併，還是無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------44. How many people received a certain survey? (1) Six-tenths of those who received the survey responded. (2) Of those who received the survey, 42 responded. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: 多少人收到調查? (1)即6/10收到調查的人會回覆時，所以無法判定，insufficient。 (2)即收到調查的人中，有42人會回覆時，所以無法判定，insufficient。 (3)由(1)(2)合併，(6/10)x=42，=> x=70，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------45. If x and z are integers, is at least one of them even? (1) x + z is odd. (2) x - z is odd. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: x, z中至少有一個為偶數? (1)即x+z為奇數時，必為一奇一偶，所以可以判定，sufficient。 (2)即x-z為奇數時，必為一奇一偶，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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46. Three thousand families live in a certain town. set?

How many families who live in the town own neither a car nor a television

(1) Of the families who live in the town, 2,980 own a car. (2) Of the families who live in the town, 2,970 own both a car and a television set. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: 多少家庭既沒車也沒電視? (A’∩B’) = (A∪B)’ = 全 - (A∪B) = 全- [A+B- (A∩B)] (1)即A=2,980時，所以無法判定，insufficient。 (2)即A∩B=2,970時，所以無法判定，insufficient。 (3)由(1)(2)合併時，不知道B，還是無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------47. 錯 In the fraction x/y, where x and y are positive integers, what is the value of y? (1) The least common denominator of x/y and 1/3 is 6. (2) x = 1. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: y=? (1) 即x/y 和 1/3 最小公分母是6時 y= 2 or 6，所以無法判定，insufficient。 (2) 即x=1時 不知道y值，所以無法判定，insufficient。 (3) 由(1)(2)合併，還是不知道y值，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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48. There are two types of rolls on a counter, plain rolls and seeded rolls. What is the total number of rolls on the counter? (1) The ratio of the number of seeded rolls on the counter to the number of plain rolls on the counter is 1 to 5. (2) There are 16 more plain rolls than seeded rolls on the counter. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: 總共有多少rolls? 問PR + SR=? (1)即SR:PR=1:5時，所以無法判定，insufficient。 (2)即PR-SR=16時，所以無法判定，insufficient。 (3)由(1)(2)合併，=> 5SR - SR=16，=>SR=4，PR=20，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------49. Is the hundredths digit of the decimal d greater than 5? (1) The tenths digit of 10d is 7. (2) The thousandths digit of d/10 is 7. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: d的百分位數大於5? (1)即10d的十分位數是7時，=>d的百分位數為7，所以可以判定，sufficient。 (2)即d/10千分位數是7時，=>d的百分位數為7，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------50. What is the value of 2x + 2y? (1)

3x + 5y = 60

(2)

5x + 3y = 68

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: 2x+2y=? (1)即3x+5y=60時，所以無法判定，insufficient。 (2)即5x+3y=68時，所以無法判定，insufficient。 (3)由(1)(2)合併，=> x=10，y=6，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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51. Whenever Martin has a restaurant bill with an amount between \$10 and \$99, he calculates the dollar amount of the tip as 2 times the tens digit of the amount of his bill. If the amount of Martin's most recent restaurant bill was between \$10 and \$99, was the tip calculated by Martin on this bill greater than 15 percent of the amount of the bill? (1) The amount of the bill was between \$15 and \$50. (2) The tip calculated by Martin was \$8. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B 【思路】題目問: 小費超過帳單的15%? (1) 即15<bill<50時 15~19 ->tips=2，佔10.5%~13.3%。 20~29 ->tips=4，佔13.8%~20%。 30~39 ->tips=6，佔15.3%~20%。 40~49 ->tips=8，佔16.3%~20%。 50 ->tips=10，佔20%。 所以無法判定，insufficient。 (2) 即tips=8時 => bill=40~49，佔16.3%~20%。 所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------52. For Manufacturer M, the cost C of producing x units of its product per month is given by C = kx + t, where C is in dollars and k and t are constants. Last month, if Manufacturer M produced 1,000 units of its product and sold all the units for k + 60 dollars each, what was Manufacturer M's gross profit on the 1,000 units? (1) Last month, Manufacturer M's revenue from the sale of the 1,000 units was \$150,000. (2) Manufacturer M's cost of producing 500 units in a month is \$45,000 less than its cost of producing 1,000 units in a month. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: 賣1,000台的毛利潤? 支出: C=k(1,000)+t 收入: (k+60)*1,000 問(k+60)*1,000 - k(1,000)+t =? (1) 即(k+60)*1,000=150,000時 => k=90，但不知道t，所以無法判定，insufficient。 (2) 即[k(1,000)+t] – [k(500)+t] =45,000時 => k(1,000 - 500)=45,000時，k=90，但不知道t，所以無法判定，insufficient。 (3)由(1)(2)合併，還是不知道t，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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53. The symbol @ represents one of the four arithmetic operations: Is (5 @ 6) @ 2 = 5 @ (6 @ 2)?

(1) 5 @ 6 = 6 @ 5 (2) 2 @ 0 = 2 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】A 【思路】題目問: (5 @ 6) @ 2 = 5 @ (6 @ 2)? (1) 即5@6=6@5時 => @可為+或*，兩者都可以成立，所以可以判定，sufficient。 (2) 即2@0=2時 => @可為+或-，可知+可以成立，-無法成立，所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------54. 錯 If y is an integer and y = |x | + x, is y = 0? (1) x < 0 (2) y < 1 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: y=0? y = |x| + x (1) 即x<0時 y= -x + x = 0，所以可以判定，sufficient。 (2) 即y<1時 因為y是整數， 又 |x|= +x (不合)或-x y只能=0，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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55. Is the positive integer n an odd integer? (1) n + 4 is a prime number. (2) n + 3 is not a prime number. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】A 【思路】題目問: 正整數n是奇數? (1) 即n+4是質數時 n必為奇數，不需要考慮n+4為質數2的特殊情形 所以可以判定，sufficient。 (2) 即n+3不是質數時 n是奇數時可以成立， 但n是偶數也有可能成立，ex: n=6,12 所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------56. What is the total surface area of rectangular solid R? (1) The surface area of one of the faces of R is 48. (2) The length of one of the edges of R is 3. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: 長方體的表面積? (1) 即僅知其中一面表面積為48時 不知道其他兩面，所以無法判定，insufficient。 (2) 即其中一邊長為3時 不知道其他兩邊長，所以無法判定，insufficient。 (3) 由(1)(2)合併，不知道其他邊長關係，還是無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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57. A certain list consists of several different integers.

Is the product of all the integers in the list positive?

(1) The product of the greatest and smallest of the integers in the list is positive. (2) There is an even number of integers in the list. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: 序列中所有整數的整數乘積是正? (1) 即序列中最大值和最小值乘積為正時 只知道為同號(+,+)(-,-)，但不知道負號中間的個數，所以無法判定，insufficient。 (2) 即序列中有偶數個整數時 但不知道數值的正負關係，所以無法判定，insufficient。 (3) 由(1)(2)合併，乘積必為正，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------58. A certain store sells chairs individually or in sets of 6. The store charges less for purchasing a set of 6 chairs than for purchasing 6 chairs individually. How much does the store charge for purchasing a set of 6 chairs? (1) The charge for purchasing a set of 6 chairs is 10 percent less than the charge for purchasing the 6 chairs individually. (2) The charge for purchasing a set of 6 chairs is \$20 more than the charge for purchasing 5 chairs individually. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: 一組6張椅子的價錢? 令零售價為x，一組6張椅子的批發價為y 已知6x>y，求y=? (1) 即y=6x*0.9=5.4x時，所以無法判定，insufficient。 (2) 即y=5x+20時，所以無法判定，insufficient。 (3) 由(1)(2)合併，=> x=50, y=270，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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59. If k

0, 1, or -1, is 1/k > 0 ?

(1) 1/(k-1) > 0 (2) 1/(k+1) > 0 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】A 【思路】題目問: 1/k>0 ? (1)即k-1>0時，=> k > 1，所以可以判定，sufficient。 (2)即k+1>0時，=> k > -1，所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------60. 錯 Does the integer k have a factor p such that 1 < p < k ? (1) k > 4! (2) 13! + 2

k

13! + 13

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B 【思路】題目問: 整數k有因數p會滿足1<p<k? (1) 即k > 4!時 可能為質數，或 k = 4!+5時 所以無法判定，insufficient。 (2) 即 13! + 2 ≦ k ≦13! + 13時 13! + 2 = 1*2*3*…*13 + 2 => 有2的因數。 13! + 13 = 1*2*3*…*13 + 13 => 有13的因數。 同理可證，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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61. When the positive integer n is divided by 25, the remainder is 13. What is the value of n? (1) n < 100 (2) When n is divided by 20, the remainder is 3. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: 正整數n=? 已知n=25a + 13，問n=? (1) 即n<100時 25a+13<100，a=0, 1, 2, 3, 4，所以無法判定，insufficient。 (2)即n=20b + 3時 =>n=100c + 63，所以無法判定，insufficient。 (3)由(1)(2)合併時，100c+63 < 100， => c=0, n=63，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------62. If y≧0, what is the value of x? (1) |x - 3| ≧ y (2) |x - 3| ≦ -y (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B 【思路】題目問: x=? 已知y≧ 0 (1) 即|x-3|≧ y時 =>|x-3|≧ 0 恆成立，所以無法判定，insufficient。 (2) 即|x-3|≦ -y時，又-y≦ 0 => |x-3|≦ 0 => |x-3|必等於0 => x-3=0，x=3，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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63. If m is a positive odd integer, what is the average (arithmetic mean) of a certain set of m integers? (1) The integers in the set are consecutive multiples of 3. (2) The median of the set of integers is 33. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: 連續m個整數的算術平均數=? 已知m為正奇數 (1) 即集合的整數是連續3的倍數時 不知道數值，所以無法判定，insufficient。 (2) 即中位數=33時 不知道其他數狀況，所以無法判定，insufficient。 (3) 由(1)(2)合併，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------64. 錯 If a < y < z < b, is |y - a| < |y - b|? (1) |z - a| < |z - b| (2) |y - a| < |z - b| (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: |y - a| < |y - b|? 直接在數線上表示判斷最快，因為絕對值就是表示距離 已知a < y < z < b，即問 y - a < - (y – b)=> 2y < a+b? (1) |z - a| < |z - b|時 即 z - a < - (z - b) => 2z < a+b 又y < z，所以可以判定，sufficient。 (2) |y - a| < |z - b|時 即 y - a < - (z - b) => y+z < a+b 又y < z，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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65. If ab ≠ 0 and points (-a, b) and (-b, a) are in the same quadrant of the xy-plane, is point (-x, y) in this same quadrant? (1) xy > 0 (2) ax > 0 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: (-x, y)也和(-a, b) (-b, a)在同一象限? 已知(-a, b) (-b, a)在同一象限，表示a, b同號 (1) 即xy>0時 表示x, y同號，但不知道x, y是否和a, b同號 所以無法判定，insufficient。 (2) 即ax>0時 表示a, x同號，但不知道y和b的關係 所以無法判定，insufficient。 (3) 由(1)(2)合併，a, b同號，x, y同號，a, x同號 => a, b, x, y同號，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------66. If q is a positive integer less than 17 and r is the remainder when 17 is divided by q, what is the value of r? (1) q > 10 (2) q = 2k, where k is a positive integer. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B 【思路】題目問: r=? 已知q<17，17=nq+r，求r=? 直接代數字判斷最快 (1) 即q>10時 q=11, 12, 13, 14, 15, 16，r=6, 5, 4, 3, 2, 1 所以無法判定，insufficient。 (2) 即q = 2k 時 q=2, 4, 8, 16，r=1, 1, 1, 1 所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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67. What is the ratio of the number of cups of flour to the number of cups of sugar required in a certain cake recipe? (1) The number of cups of flour required in the recipe is 250 percent of the number of cups of sugar required in the recipe. (2) 1.5 more cups of flour than cups of sugar are required in the recipe. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】A 【思路】題目問: 麵粉和糖的比例? (1)即F:S = 2.5:1時，所以可以判定，sufficient。 (2)即F:S= 1.5+s:s時，所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------68. In 1999 Company X's gross profit was what percent of its revenue? (1) In 1999 Company X’s gross profit was 1/3 of its expenses. (2) In 1999 Company X’s expenses were 3/4 of its revenue. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: G:R=? 已知R – E = G (1)即G=(1/3)E時，=> R=(4/3)E，=> G:R=1:4，所以可以判定，sufficient。 (2)即E=(3/4)R時，=> G=(1/4)R，=> G:R=1:4，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------69. At the bakery, Lew spent a total of \$6.00 for one kind of cupcake and one kind of doughnut. How many doughnuts did he buy? (1) The price of 2 doughnuts was \$0.10 less than the price of 3 cupcakes. (2) The average (arithmetic mean) price of 1 doughnut and 1 cupcake was \$0.35. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: 買了多少甜甜圈? 已知mc+nd=6，問n=? (1)即2d=3c-0.1時，所以無法判定，insufficient。 (2)即d+c=0.35時，所以無法判定，insufficient。 (3)由(1)(2)合併，c=0.12, d=0.23，還是無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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70. If eleven consecutive integers are listed from least to greatest, what is the average (arithmetic mean) of the eleven integers? (1) The average of the first nine integers is 7. (2) The average of the last nine integers is 9. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: 11個連續整數的算術平均值? 已知是由小到大排列，公差為1，問平均值即a6=? (1) 即前9個數平均為7時 => a5=7， => a6=8 所以可以判定，sufficient。 (2) 即後9個數平均為7時 => a7=9， => a6=8 所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------71. What is the greatest integer that is less than t? (1) t = 9/4 (2) t = (-3/2)2 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: 小於t的最大整數? (1) 即t=9/4=2.25時，所求=2，所以可以判定，sufficient。 (2) 即t=9/4=2.25時，所求=2，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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72. Of the 4,800 voters who voted for or against Resolution K, 1800 were Democrats and 3,000 were Republicans. What was the total number of female voters who voted for Resolution K? (1) 3/4 of the Democrats and 2/3 of the Republicans voted for Resolution K. (2) 1/3 of the Democrats who voted for Resolution K and 1/2 of the Republicans who voted for Resolution K were females. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: 多少女性投票支持K法案? (1) 即3/4民主黨員和2/3共和黨員支持法案時 只知道支持人數有(3/4*1,800)+(2/3*3,000) 但不知道女性比例，所以無法判定，insufficient。 (2) 即支持法案的民主黨員有1/3是女性和支持法案的共和黨員有1/2是女性時 不知道詳細支持人數，所以無法判定，insufficient。 (3) 由(1)(2)合併時，(3/4*1,800*1/3)+ (2/3*3,000*1/2)=1,450 所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------73. If n is a positive integer, is n3 - n divisible by 4? (1) n = 2k + 1, where k is an integer. (2) n2 + n is divisible by 6. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】A 【思路】題目問: n3 - n 可以被4整除? (1) 即n=2k+1時 n3 – n=(2k+1)3 - (2k+1) = (2k+1)(4k2+4k)=4k(k+1)(2k+1) 所以可以判定，sufficient。 (2) 即n2+n=n(n+1)=6a時 n3 – n=n(n+1)(n-1)=6a(n-1) 所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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74. If x ≠ -y, is (x-y)/(x+y) > 1? (1) x > 0 (2) y < 0 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: (x-y)/(x+y) > 1? 已知(x-y)/(x+y) = (x+y-2y)/(x+y) = 1 - [2y/(x+y)] 問1-[2y/(x+y)] >1 => 即判斷2y/(x+y) < 0? (1)即x>0時，不知道y的正負，所以無法判定，insufficient。 (2)即y<0時，不知道x的正負，所以無法判定，insufficient。 (3)由(1)(2)合併時，不知道x+y的正負，所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------75. A certain group of car dealerships agreed to donate x dollars to a Red Cross chapter for each car sold during a 30-day period. What was the total amount that was expected to be donated? (1) A total of 500 cars were expected to be sold. (2) 60 more cars were sold than expected, so that the total amount actually donated was \$28,000. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: 預期總捐贈金額=? 問nx=? (1)即n=500時，不知道x，所以無法判定，insufficient。 (2)即(n+60)x=28,000時，不知道x，所以無法判定，insufficient。 (3)由(1)(2)合併，(500+60)x=28,000，=> x=500，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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76. Over a certain time period, did the number of shares of stock in Ruth's portfolio increase? (1) Over the time period, the ratio of the number of shares of stock to the total number of shares of stocks and bonds in Ruth's portfolio increased. (2) Over the time period, the total number of shares of stocks and bonds in Ruth's portfolio increased. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: 在投資組合中股份會增加? (1) 即S: S+B增加時 可能S增加或B減少，所以無法判定，insufficient。 (2) 即S+B增加時 可能S增加或B增加，所以無法判定，insufficient。 (3)由(1)(2)合併，S必增加，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------77. 錯 Is |x| = y - z? (1) x + y = z (2) x < 0 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: |x| = y - z? (1) 即x+y=z時 x=z–y 但不知道x的正負，|x| = z - y或y – z 所以無法判定，insufficient。 (2) 即x<0時，不知道y, z，所以無法判定，insufficient。 (3) 由(1)(2)合併時，x<0 => |x| = y - z，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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78. If n is an integer, is -3xn positive? (1) x is negative. (2) n is odd. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: -3xn > 0? (1)即x<0時，不知道n的個數，所以無法判定，insufficient。 (2)即n是奇數時，不知道x的正負，所以無法判定，insufficient。 (3)由(1)(2)合併，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------79. If n is an integer between 2 and 100 and if n is also the square of an integer, what is the value of n? (1) n is even. (2) The cube root of n is an integer. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B 【思路】題目問: n=? 已知2<n<100，n=x2，問n=? (1) 即n是偶數時 n= 22, 42, 62, 82，所以無法判定，insufficient。 (2) 即n的立方根是整數時 n可能為23(不合), 33(不合), 43 所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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80. 錯 If x and y are positive integers such that x = 8y + 12, what is the greatest common divisor of x and y? (1) x = 12u, where u is an integer. (2) y = 12z, where z is an integer. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B 【思路】題目問x和y的最大公因數? (1) 即x=12u時 => 12 u = 8y + 12 => 8y是12的倍數 => y至少包含一個3 (x, y)可能為3, 6, 12，所以無法判定，insufficient。 (2) 即y=12z時 => x = 8 (12z) + 12 => x = 12 (8z+1) (x, y)=12，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------81. At a certain bakery, each roll costs r cents and each doughnut costs d cents. If Alfredo bought rolls and doughnuts at the bakery, how many cents did he pay for each roll? (1) Alfredo paid \$5.00 for 8 rolls and 6 doughnuts. (2) Alfredo would have paid \$10.00 if he had bought 16 rolls and 12 doughnuts. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: 對每根蛋捲付多少錢? 問r=? (1)即8r+6d=5時，所以無法判定，insufficient。 (2)即16r+12d=10時，所以無法判定，insufficient。 (3)由(1)(2)合併，還是無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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82. Is x > 0? (1) xy > 0 (2) x + y > 0 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: x>0? (1)即xy>0時，不知道y，所以無法判定，insufficient。 (2)即x+y>0時，不知道y，所以無法判定，insufficient。 (3)由(1)(2)合併，(x, y)=(+,+)，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------83. How much time did it take a certain car to travel 400 kilometers? (1) The car traveled the first 200 kilometers in 2.5 hours. (2) If the car's average speed had been 20 kilometers per hour greater than it was, it would have traveled the 400 kilometers in 1 hour less time than it did. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B 【思路】題目問: 400km的旅程需要多少時間? (1) 即前200km要花2.5hr時 不知道後200km關係，所以無法判定，insufficient。 (2) 即400/v – 400/(v+20) = 1時 =>v=80，400/80=5，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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84. Each of the offices in a certain building has a floor area of 200, 300, or 350 square feet. How many offices are on the first floor of the building? (1) There is a total of 9,500 square feet of office floor space on the first floor of the building. (2) Ten of the offices on the first floor have floor areas of 350 square feet each. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: 一樓有多少間辦公室? 令200平方呎有x間，300平方呎有y間，350平方呎有z間，問x+y+z=? (1)即200x+300y+350z=9,500時，所以無法判定，insufficient。 (2)即z=10時，所以無法判定，insufficient。 (3)由(1)(2)合併，還是無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------85. Is n divisible by 12? (1) n/6 is an integer. (2) n/4 is an integer. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: n可否被12整除? (1) 即n可以被6整除時 => n=6a，n=6, 12，所以無法判定，insufficient。 (2) 即n可以被4整除時 => n=4b時，n=4, 8, 12，所以無法判定，insufficient。 (3)由(1)(2)合併，n=12c，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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86. Do more than 50 percent of the children in a certain group have brown hair? (1) 70 percent of the boys in the group have brown hair. (2) 30 percent of the children in the group are girls with brown hair. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: 超過50%的孩子有棕髮? (1) 即70%男生是棕髮時，不知道女生狀況，所以無法判定，insufficient。 (2) 即30%女生是棕髮時，不知道男生狀況，所以無法判定，insufficient。 (3) 由(1)(2)合併，不知道男女人數狀況，還是無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------87. Is the integer n a multiple of 15? (1) n is a multiple of 20. (2) n + 6 is a multiple of 3. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: n是15的倍數? (1)即n=20a時，所以無法判定，insufficient。 (2)即n+6=3b時，n=3b-6，所以無法判定，insufficient。 (3)由(1)(2)合併，n=60c+60，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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88. If x + y ≠ 0, what is the value of (ax+ay)/(x+y)? (1) x = 4 and y = 5. (2) a = 6 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B 【思路】題目問: (ax+ay)/(x+y)=? 化簡後，即問a=? (1)即x=4, y=5時，所以無法判定，insufficient。 (2)即a=6時，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------89. What is the ratio of p to r? (1) The ratio of p to 3r is 5 to 9. (2) The sum of p and r is 16. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】A 【思路】題目問: p:r=? (1)即p:3r=5:9時，=> p:r=5:3，所以可以判定，sufficient。 (2)即p+r=16時，所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------90. What is the value of a4 - b4? (1) a2 - b2 = 16 (2) a + b = 8 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: a4 - b4=? 即(a2+b2)(a2-b2)=? (1) 即a2 - b2=16時 不知道a2+b2，所以無法判定，insufficient。 (2) 即a+b=8時 (a2+b2)(a2-b2)= (a2+b2)(a+b)(a-b) 不知道(a2+b2)和(a-b)，所以無法判定，insufficient。 (3)由(1)(2)合併，a-b=2 => a=5, b=3，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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91. In the rectangular coordinate system, are the points (r, s) and (u, v) equidistant from the origin? (1) r + s = 1 (2) u = 1 - r and v = 1 - s. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: (r,s)和(u,v)到原點距離相同? 問r2+s2=u2+v2? (1) 即r+s=1時 不知道u, v，所以無法判定，insufficient。 (2) 即u=1-r和v=1-s時 u2=r2-2r+1，v2=s2-2s+1 u2+ v2= r2+s2-2(r+s)+2， 但不知道r+s，所以無法判定，insufficient。 (3)由(1)(2)合併，r2+s2=u2+v2，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------92. X and Y are sets of positive integers. Is the greatest integer in X greater than the greatest integer in Y? (1) X is a set of 5 consecutive odd integers, each less than 20. (2) Y is a set of 3 consecutive even integers, each less than 15. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: 集合X的最大整數會大於集合Y的最大整數? (1)即集合X為連續5個正奇數，且各數都小於20時 集合X的最大整數可能為9, 11, 13, 15, 17, 19…等 但不知道集合Y的情形，所以無法判定，insufficient。 (2)即集合Y為連續3個正偶數，且各數都小於15時 集合Y的最大整數可能為6, 8, 10, 12, 14…等 但不知道集合X的情形，所以無法判定，insufficient。 (3)由(1)(2)合併，還是無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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93. The lifetimes of all the batteries produced by a certain company in a year have a distribution that is symmetric about the mean m. If the distribution has a standard deviation of d, what percent of the distribution is greater than m + d? (1) 68 percent of the distribution lies in the interval from m - d to m + d, inclusive. (2) 16 percent of the distribution is less than m - d. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: 有幾成的分布會大於m+d? 已知這個常態分布和平均值成對稱 (1)即m-d< x <m+d佔了68%時: 小於m-d的機率=大於m+d的機率= (1 - 68%) / 2 = 16%，所以可以判定，sufficient。 (2)即16%的機率小於m-d時: 同理，大於m+d的機率=16%，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------94. If |x + 2| = 4, what is the value of x? (1) x2 ≠ 4 (2) x2 = 36 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: x=? 已知|x + 2|=4，=> x=2或-6 (1) 即x2≠4時，=> x=-6，所以可以判定，sufficient。 (2) 即x2=36時，=> x=-6，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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95. If v and w are different integers, does v = 0? (1) vw= v2 (2) w = 2 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】A 【思路】題目問: v=0? 已知v≠w (1) 即vw=v2時，v(v-w)=0，所以v=0或v=w(不合) 所以可以判定，sufficient。 (2) 即w=2時，只知道v≠2， 所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------96. Is √(x-5)2 = 5 - x (1) -x|x| > 0 (2) 5 - x > 0 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: √(x-5)2 = 5 - x? 表示 x-5 < 0時才會成立，也就是x< 5才會成立 (1) 即-x|x|>0時，=> -x >0 => x<0時 所以可以判定，sufficient。 (2) 即5-x > 0時，=> x<5，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------97. What is the median of a certain set of 7 numbers? (1) 3 of the numbers are less than 10. (2) 4 of the numbers are greater than 10. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: 7個數字的集合其中位數為? 問a4=? (1)即3個數字小於10時，不知道a4，所以無法判定，insufficient。 (2)即4個數字大於10時，只知道a4>10，所以無法判定，insufficient。 (3)由(1)(2)合併，還是無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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98. Is x + y < 1? (1) x < 8/9 (2) y < 1/8 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: x + y < 1? (1) 即x < 8/9時，不知道y，所以無法判定，insufficient。 (2) 即y < 1/8時，不知道x，所以無法判定，insufficient。 (3)由(1)(2)合併時，x+y < 73/72，所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------99. For each home sold in County X, the buyer and the seller each must pay to County X a tax of 0.5 percent of the sale price of the home. Colleen recently sold her old home and bought a new home, both in County X. What was the total tax that Colleen paid to County X on these home sales? (1)

Colleen's old home had a sale price of \$169,500.

(2)

Colleen's new home had a sale price 20% greater than that of her old home.

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: 總共要付多少房屋稅? 即x*0.5%+y*0.5%=? (1)即x=169,500時，但不知道y，所以無法判定，insufficient。 (2)即y=1.2x時，但不知道x，所以無法判定，insufficient。 (3)由(1)(2)合併，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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100. A certain bank charges a maintenance fee on a standard checking account each month that the balance falls below \$1,000 at any time during the month. Did the bank charge a maintenance fee on Sue's standard checking account last month? (1) At the beginning of last month, Sue's account balance was \$1,500. (2) During last month, a total of \$2,000 was withdrawn from Sue's checking account. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: 銀行上個月有向Sue收費? 已知只要存款餘額低於\$1000就要收費。 (1)即上個月一開始餘額有1,500時，但不知道中間帳戶餘額情形，所以無法判定，insufficient。 (2)即上個月總共領出2,000時，但不知道帳戶餘額情形，所以無法判定，insufficient。 (3)由(1)(2)合併，還是無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------101.

In the figure above, lines l1 and l2 are parallel. What is the value of x? (1) y = 87 (2) z = 93 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: x=? (1)即y=87時，=> x=93，所以可以判定，sufficient。 (2)即z=93時，=> x=93，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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102.

In the coordinate plane above, what is the area of rectangular region MNRS? (1) M has coordinates (2, 1) (2) N has coordinates (2, 5) (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: MNRS的面積? (1)即M(2, 1)時，無法確定長寬，所以無法判定，insufficient。 (2)即N(2, 5)時，無法確定長寬，所以無法判定，insufficient。 (3)由(1)(2)合併，只能確定寬，無法確定長，還是無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------103. How much did the taxi driver charge George for the trip to the airport? (1) George paid the taxi driver a tip equal to 15 percent of the amount the driver charged. (2) George paid the taxi driver a tip of \$6.00. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: 計程車費收多少? 問T=? (1)即tip=0.15T時，但不知道T，所以無法判定，insufficient。 (2)即tip=6時，但不知道T，所以無法判定，insufficient。 (3)由(1)(2)合併，可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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104. A child selected a three-digit number, XYZ, where X, Y, and Z denote the digits of the number. If no two of the three digits were equal, what was the three-digit number? (1) The sum of the digits was 10. (2) X < Y < Z (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: 三位數xyz=? 已知x, y, z相異 (1)即x+y+z=10時，所以無法判定，insufficient。 (2)即x<y<z時，所以無法判定，insufficient。 (3)由(1)(2)合併，還是無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------105. If b is positive, is ab positive? (1) a2 b > 0 (2) a2 + b = 13 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: ab>0? 已知b>0 (1) 即 a2 b > 0時，=> b>0，但a可能為正或負，所以無法判定，insufficient。 (2) 即 a2 + b = 13時，但a可能為正或負，所以無法判定，insufficient。 (3) 由(1)(2)合併，還是無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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106. Robin split a total of \$24,000 between two investments, X and Y. If investment Y earns 7 percent simple annual interest, how much of the total did Robin put into investment Y? (1) Each investment earns the same dollar amount of interest annually. (2) Investment X earns 5 percent simple annual interest. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: 多少資金分配到投資案Y? 已知投資案X利息: (24,000-y)*Ra，投資案Y利息: y*7%，問y=? (1)即(24,000-y)*Ra=y*7%時，所以無法判定，insufficient。 (2)即Ra=5%時，所以無法判定，insufficient。 (3)由(1)(2)合併時，(24,000-y)*5=y*7，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------107. 錯 Is 2x - 3y < x2? (1) 2x - 3y = -2 (2) x > 2 and y > 0 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: 2x - 3y < x2? 問x 2-2x+3y> 0 =>問 x(x-2) +3y > 0? (1) 即2x -3y=-2時 原式恆成立，所以可以判定，sufficient。 (2) 即x>2且y>0時 (解1): 原式恆成立，所以可以判定，sufficient。 (解2): x >2 => x2 > 2x 又 y >0，x2 > 2x - 3y，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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108. If x/2 = 3/y, is x less than y? (1) y ≧ 3 (2) y ≦ 4 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】A 【思路】題目問: x < y? 已知xy=6 (1) 即y≧3時，=> x≦2，所以可以判定，sufficient。 (2) 即y≦4時，=> x≧1.5，所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------109. Marta bought several pencils. If each pencil was either a 23-cent pencil or a 21-cent pencil, how many 23-cent pencils did Marta buy? (1) Marta bought a total of 6 pencils. (2) The total value of the pencils Marta bought was 130 cents. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B 【思路】題目問: 買了多少23-cent鉛筆? 令23-cent鉛筆為x，21-cent鉛筆為y，問x=? (1) 即x+y=6時 所以無法判定，insufficient。 (2) 即23x+21y=130時 因為(23,21)|130，所以必有整數解。 代入整數解檢驗 => (x, y) =(2, 4) 所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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110. If x and y are integers and x > 0, is y > 0? (1) 7x - 2y > 0 (2) -y < x (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: y > 0? 已知x > 0 (1) 即7x - 2y > 0時 => 7x > 2y，y可以為正或負 所以無法判定，insufficient。 (2) 即-y < x時 => y > -x，y可以為正或負 所以無法判定，insufficient。 (3) 由(1)(2)合併，-x < y < 7x/2 y可以為正或負，還是無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------111. A state legislature had a total of 96 members. The members who did not vote on a certain bill consisted of 25 who were absent and 3 who abstained. How many of those voting voted for the bill? (1) Exactly 1/3 of the total membership of the legislature voted against the bill. (2) The number of Legislators who voted for the bill was 8 more than the total number who were absent or abstained. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: 多少支持法案通過的票? 已知支持或反對人數: 96-25-3=68=Y+N，問Y=? (1)即N=1/3 *96=32時，所以可以判定，sufficient。 (2)即Y=(25+3)+8=36時，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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112. If m and n are positive integers and mn = k, is m + n = k + 1? (1) m = 1 (2) k is a prime number. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: m+n = k+1? 已知mn = k (1)即m=1時，n=k，所以可以判定，sufficient。 (2)即k是質數時，(m,n)必為1和本身= (1,k)或(k,1)，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------113. In a certain year the United Nations' total expenditures were \$1.6 billion. Of this amount, 67.8 percent was paid by the 6 highest-contributing countries, and the balance was paid by the remaining 153 countries. Was Country X among the 6 highest-contributing countries? (1) 56 percent of the total expenditures was paid by the 4 highest-contributing countries, each of which paid more than Country X. (2) Country X paid 4.8 percent of the total expenditures. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: Country X是六大金主之一嗎? (1)不知道Country X是否在另外兩國之內，所以無法判定，insufficient。 (2)Country X付出4.8%費用時，還是無法判定，insufficient。 (3)由(1)(2)合併，不知道X是否在另外兩國之內，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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114. If (x+y)/z = -2, is x positive? (1) z is negative. (2) y is positive. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: x>0? 已知(x+y)/z = -2，=> x+y=-2z，問x>0? (1)即z<0時，x=-(y+2z)，不知道y，所以無法判定，insufficient。 (2)即y>0時，x=-(y+2z)，不知道z，所以無法判定，insufficient。 (3)由(1)(2)合併，不知道y+2z的正負，所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------115. 錯 List S and list T each contain 5 positive integers, and for each list the average (arithmetic mean) of the integers in the list is 40. If the integers 30, 40, and 50 are in both lists, is the standard deviation of the integers in list S greater than the standard deviation of the integers in list T? (1) The integer 25 is in list S. (2) The integer 45 is in list T. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: 標準差S大於標準差T? The main idea is to think of SD as of average deviation from mean:

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116. On the number line, what is the distance between the point 2x and the point 3x? (1) On the number line, the distance between the point -x and the point x is 16. (2) On the number line, the distance between the point x and the point 3x is 16. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: 數線上2x到3x的距離=? 問|3x-2x|=|x|=? (1)即|x - (-x)|=16時，2|x|=16，|x|=8，所以可以判定，sufficient。 (2)即|3x - x|=16時，2|x|=16，|x|=8，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------117. What is the remainder when the positive integer n is divided by the positive integer k, where k > 1? (1) n = (k + 1)3 (2) k = 5 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】A 【思路】題目問: n除以k的餘數=? (1) 即n=(k + 1)3時 n=k3+3k2+3k+1=k(k2+3k+3)+1 => r=1，所以可以判定，sufficient。 (2) 即k=5時，但不知道n，所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------118. Is x ≧ 3? (1) x2 – 9 = 0 (2) x < 10 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: x≧3? (1)即x=3或-3時，所以無法判定，insufficient。 (2)即x<10時，所以無法判定，insufficient。 (3)由(1)(2)合併，還是無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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119. At a two-candidate election for mayor, 3/4 of the registered voters cast ballots. the winning candidate?

How many registered voters cast ballots for

(1) 25,000 registered voters did not cast ballots in the election. (2) Of the registered voters who cast ballots, 55 percent cast ballots for the winning candidate. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: 多少人投票支持當選人? 已知3/4選民投票 (1)即25,000人沒投票時，可知道選民總人數=25,000÷(1/4)=100,000，所以無法判定，insufficient。 (2)即投票的人中55%支持當選人時，只知比例，不知道人數，所以無法判定，insufficient。 (3)由(1)(2)合併，100,000*(3/4)*55%=41,250，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------120. If n is the product of the least and the greatest of 6 consecutive integers, what is the value of n? (1) The greatest of the 6 consecutive integers is 20. (2) The average (arithmetic mean) of the 6 consecutive integers is 17.5. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: n=? 已知n是6個連續整數中最大值和最小值的成績 設a, a+1, a+2, a+3, a+4, a+5，求a(a+5)=? (1)即a+5=20時，=>a=15，所以可以判定，sufficient。 (2)即a+2.5=17.5時，=>a=15，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------121. If n and s are each 2-digit positive integers, is n greater than s? (1) The units digit of n is greater than the units digit of s. (2) The tens digit of n is greater than the tens digit of s. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B 【思路】題目問: n>s? 令n=ab，s=mn (1)即b>n時，不知道a,m關係，所以無法判定，insufficient。 (2)即a>m時，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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122. 錯 If n is a positive integer and r is the remainder when (n - 1)(n + 1) is divided by 24, what is the value of r? (1) 2 is not a factor of n. (2) 3 is not a factor of n. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: r=? 有時候直接代入數字會比較容易幫助判斷 (1)即n=2a+1時 (n-1)(n+1)=2a (2a+2)=4a(a+1) a=1 => (n-1)(n+1)=8，=> r=8 a=2 => (n-1)(n+1)=24，=> r=0 所以無法判定，insufficient。 (2)即n=3b+1或3b+2時 (i)(n-1)(n+1)=3b (3b+2) b=1 => (n-1)(n+1)=3*5，=> r=15 b=2 => (n-1)(n+1)=6*8，=> r=0 (ii)(n-1)(n+1)=(3b+1) (3b+3)=3(b+1)(3b+1) b=1 => (n-1)(n+1)=3*2*4=24，=> r=0 b=2 => (n-1)(n+1)=3*3*7=63，=> r=15 所以無法判定，insufficient。 (3)由(1)(2)合併時，即n=6c+1或6c+5 (i)(n-1)(n+1)=6c (6c+2)=12c(3c+1) c=1 => (n-1)(n+1)=12*4=48，=> r=0 c=2 => (n-1)(n+1)=24*7，=> r=0 (ii)(n-1)(n+1)=(6c+4) (6c+6)=12(3c+2)(c+1) c=1 => (n-1)(n+1)=12*5*2，=> r=0 c=2 => (n-1)(n+1)=12*8*3，=> r=0 所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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123. When positive integer n is divided by 3, the remainder is 2; and when positive integer t is divided by 5, the remainder is 3. What is the remainder when the product nt is divided by 15? (1) n - 2 is divisible by 5. (2) t is divisible by 3. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: nt除以15的餘數=? 已知n=3a+2, t=5b+3 (1) 即n-2 = 5c時 n = 5c+2 = 3a+2時，=>n=15m+2 nt=(15m+2)(5b+3)=75mb+45m+10b+6=15(5mb+3m)+10b+6 所以無法判定，insufficient。 (2) 即t=3d=5b+3時，=> t=15n+3 nt=(3a+2)(15n+3)=45na+9a+30n+6=15(3na+2n)+9a+6 所以無法判定，insufficient。 (3) 由(1)(2)合併，nt=(15m+2)(15n+3)=15(15mn+3m+2n)+6， r=6，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------124. A certain motel has a total of 540 units, each of which has a 1-person, 2-person, or 4-person capacity. How many people stayed in the motel's 4-person units yesterday? (1) At this motel, 1/3 of the units are 4-person units. (2) Yesterday, 80 percent of the 4-person units in the motel were filled to capacity, and the rest of the 4-person units were empty. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: 昨天四人房住了多少人? 已知單人房有x間，雙人房有y間，四人房有z間，x+y+z=540 (1) 即z=540*(1/3)=180時 但不知道四人房住房率，所以無法判定，insufficient。 (2) 即昨日四人房住房率為80%時 但不知道四人房有多少間，所以無法判定，insufficient。 (3)由(1)(2)合併，180*80%=144(間)，所求=144*4=576，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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125. If @ denotes one of two arithmetic operations, addition or multiplication, and if k is an integer, what is the value of 3 @ k? (1) 2 @ k = 3 (2) 1 @ 0 = k (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】A 【思路】題目問: 3 @ k=? 已知k為整數 (1) 即2@k=3時 (i)@為+時，=> k=1，=> 所求= 3+1=4 (ii)@為* => k=3/2(不合) 所以可以判定，sufficient。 (2) 即1@0=k時 (i)@為+時，=> k=1，=> 所求= 3+1=4 (ii)@為* => k=0，=> 所求= 3*0=0 所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------126. If r and t are positive integers, is rt even? (1) r + t is odd. (2) rt is odd. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】A 【思路】題目問: rt是偶數? (1)即r+t是奇數時，(r,t)可能為(奇,偶)或(偶,奇)，所以可以判定，sufficient。 (2)即rt是奇數時，可知r必為奇數，但不知道t，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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127. 錯 If 2/5 of the students at College C are business majors, what is the number of female students at College C? (1) 2/5 of the male students at College C are business majors. (2) 200 of the female students at College C are business majors. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: 大學裡有多少女生? 已知(2/5)(M+F)主修企管，問F=? (1) 即(2/5)M主修企管時 不知道人數，所以無法判定，insufficient。 (2) 即200位女性學生主修企管時 不知道其他女性人數，所以無法判定，insufficient。 (3) 由(1)(2)合併，(2/5)(M)+200=(2/5)(M+F) (2/5)(F) = 200，F=500 所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------128. Is y < (x+z)/2? (1) y - x < z - y (2) z - y < (z-x)/2 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: y < (x+z)/2 ? (1) 即y - x < z - y時，=> y < (x+z)/2，所以可以判定，sufficient。 (2) 即z - y < (z-x)/2時，=> y > (x+z)/2，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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129. 錯 If t is a positive integer and r is the remainder when t2 + 5t + 6 is divided by 7, what is the value of r? (1) When t is divided by 7, the remainder is 6. (2) When t2 is divided by 7, the remainder is 1. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】A 【思路】題目問: r=? (1) 即t=7a+6時 t2+5t+6=(t+2)(t+3)=(7a+8)(7a+9) r=72除以7的餘數=2，所以可以判定，sufficient。 (2) 即t2=7b+1時，t2+5t+6=7b+1+5t+6=7(7b+1)+5t， (i)t=1 => r=5 (ii)t=6 => r=2 (iii)t=8 => r=5 所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------130. Set S consists of 20 different positive integers. How many of the integers in S are odd? (1) 10 of the integers in S are even. (2) 10 of the integers in S are multiples of 4. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】A 【思路】題目問: 集合S有多少個奇數? 已知集合S包含20個不同正整數 (1) 即集合S有10個偶數時 => 表示必有10個奇數，所以可以判定，sufficient。 (2) 即集合S有10個4的倍數時 => 無法推知偶數到底有多少個，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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131. 錯. If w + x < 0, is w - y > 0? (1) x + y < 0 (2) y < x < w (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B 【思路】題目問: w - y > 0? 已知w+x<0，問w>y? 注意，兩個不等式只能相加，不能相減，因為可能會有變號問題 (1) 即x + y < 0時 兩式相加，=> w+2x+y<0，=> w+y<-2x 所以無法判定，insufficient。 (2) 即y < x < w時 所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------132. 錯 If the drama club and music club are combined, what percent of the combined membership will be male? (1) Of the 16 members of the drama club, 15 are male. (2) Of the 20 members of the music club, 10 are male. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: 社團合併後的男生比率? 所求=(DM+MM)/(DM+DF)+(MM+MF) (1)即drama club中的16個會員有15位是男性時 可知有1位是女性，但不知道drama club總人數 所以無法判定，insufficient。 (2)即music club中的20個會員有10位是男性時 可知有10位是女性，但不知道music club總人數 所以無法判定，insufficient。 (3)由(1)(2)合併，還是不知道總人數，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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133. What is the average (arithmetic mean) height of the n people in a certain group? (1) The average height of the n/3 tallest people in the group is 6 feet 2 1/2 inches, and the average height of the rest of the people in the group is 5 feet 10 inches. (2) The sum of the heights of the n people is 178 feet 9 inches. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】A 【思路】題目問: 身高的平均值? (1)所求=(6 ft 2.5 in)*[(n/3)/n]+ (5 ft 10 in)*[(2/3)n/n]，所以可以判定，sufficient。 (2)但不知道n，所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------134. If m and n are integers, is m odd? (1) n + m is odd. (2) n + m = n2 + 5. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B 【思路】題目問: m是奇數? 直接代入數字判斷最快 (1) 即n+m是奇數時 不知道n，所以無法判定，insufficient。 (2) 即n+m=n2+5時 => m =n2 - n + 5 n=奇數時 = > m為奇數 n=偶數時 = > m為奇數 所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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135. How many more first-time jobless claims were filed in week P than in week T? (1) For weeks P, Q, R, and S, the average (arithmetic mean) number of first-time jobless claims filed was 388,250. (2) For weeks Q, R, S, and T, the average (arithmetic mean) number of first-time jobless claims filed was 383,000. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: Week P比Week T多了多少? (1) 即Week P+ Week Q+ Week R+ Week S = 388,250*4時 不知道Week T，所以無法判定，insufficient。 (2) 即Week Q+ Week R+ Week S+ Week T = 383,000*4時 不知道Week P，所以無法判定，insufficient。 (3)由(1)(2)合併，Week P - Week T=5,250*4，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------136. 錯 x, 3, 1, 12, 8 If x is an integer, is the median of the 5 numbers shown greater than the average (arithmetic mean) of the 5 numbers? (1) x > 6 (2) x is greater than the median of the 5 numbers. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: 中位數大於平均值? 已知1, 3, 8, 12, x，median可能為3, x, 8，average = (24+x)/5 (1) 即x>6時 (i) x = 7, then the list is 1, 3, 7, 8, 12, median = 7，average = 31/5 = 6.2，answer to question is YES (ii) x = 8, then the list is 1, 3, 8, 8, 12, median = 8，average = 32/5 = 6.4，answer to question is YES (iii) x =極大時，then the list is 1, 3, 8, 12, x, median = 8，average = 極大，answer to question is No 所以無法判定，insufficient。 (2) 即x>中位數時，中位數必為8，所以x>8 (i) The list is 1, 3, 8, x, 12, 即9≦x≦12時 median = 8，average = (24+x)/5，6.6≦avg≦7.2，answer to question is No (ii) The list is 1, 3, 8, 12, x, 即x≧12時 median = 8，average = (24+x)/5，avg≧7.2，不一定成立。 所以無法判定，insufficient。 (3)由(1)(2)合併，還是無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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137. 錯 If x and y are positive integers, is xy a multiple of 8? (1) The greatest common divisor of x and y is 10. (2) The least common multiple of x and y is 100. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: xy是8的倍數? 已知8=23，也就是含有三個2 (1) 即(x,y)=10=d時，x=10h, y=10k，xy=102hk=22*52*hk，表示x或y都含有一個2 不知道h或k是否還有2來湊成三個2 所以無法判定，insufficient。 (2) 即[x,y]=100=22*52=dhk時，只能表示x或y至少有一含有兩個2 不知道另一個是否還有2來湊成三個2 所以無法判定，insufficient。 (3) 由(1)(2)合併，xy=(x,y)*[x,y]=1000=23*53 所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------138. 錯 If zy < xy < 0, is |x - z| + |x| = |z|? (1) z < x (2) y > 0 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: |x - z| + |x| = |z|? 已知zy < xy < 0 (1)即z < x時 =>可知 y > 0，不然不等式方向會改變 =>|x - z| + |x|= (x-z) + (-x) = -z = |z| 所以可以判定，sufficient。 (2)即y > 0時 =>可知 z < x，不然不等式方向會改變 => |x - z| + |x|= (x-z) + (-x) = -z = |z| --------------------------------------------------------------------------------------------------------------------------------------------------------

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139. If machine J, working alone at its constant rate, takes 2 minutes to wrap 60 pieces of candy, how many minutes does it take machine K, working alone at its constant rate, to wrap 120 pieces of candy? (1) Machine K, working alone at its constant rate, takes more than 5 minutes to wrap 60 pieces of candy. (2) Machines J and K, working together at their respective constant rates, take 1 minute and 30 seconds to wrap 60 pieces of candy. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B 【思路】題目問: k機器包120顆糖果要多少時間? (1) 即k機器用超過5分鐘包60顆糖果時 所以無法判定，insufficient。 (2) 即60/2 + Vk = 60/1.5 時 Vk=10，tk= 120/Vk=12(mins) 所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------140. 錯 A manufacturer produced x percent more video cameras in 1994 than in 1993 and y percent more video cameras in 1995 than in 1994. If the manufacturer produced 1,000 video cameras in 1993, how many video cameras did the manufacturer produce in 1995? (1) xy = 20 (2) x + y + xy/100 = 9.2 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B 【思路】題目問: 1995年生產多少台攝影機? 已知[f(1994)-f(1993)]/f(1993)=x/100 [f(1995)-f(1994)]/f(1994)=y/100 又f(1993)=1,000，f(1994)=1,000+10x， f(1995)=1,000+10x+[1,000+10x]*(y/100)=1,000+10x+10y+(xy/10)，問f(1995)=? (1) 即xy=20時 不知道x, y，所以無法判定，insufficient。 (2) 即x + y + xy/100 = 9.2時 f(1995)=1,000+10[x+y+(xy/100)]=1,092 所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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141. Of the 20 people who each purchased 2 tickets to a concert, some used both tickets, some used only 1 ticket, and some used neither ticket. What percent of the tickets that were purchased by the 20 people were used by those people? (1) Of the 20 people, 10 used only 1 ticket. (2) Of the 20 people, 4 used neither ticket. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: 買來的票被用掉多少比率? (1) 即在20個人中有10個人只用掉一張票時 但不知道其他人的使用狀況，所以無法判定，insufficient。 (2) 即在20個人中有4個人沒用掉票時 不知道其他人的使用狀況，所以無法判定，insufficient。 (3) 由(1)(2)合併，可知剩下6個人用掉兩張票， 所以(6*2+10*1)/(20*2)=55%，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------142. At a certain stand, all soft drinks cost the same and all sandwiches cost the same. How much does 1 sandwich cost at the stand? (1) At the stand, 1 sandwich and 2 soft drinks cost a total of \$3.15. (2) At the stand, 3 sandwiches and 1 soft drink cost a total of \$5.70. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: 一個三明治要多少錢? 問s=? (1)即s+2d=3.15時，不知道d，所以無法判定，insufficient。 (2)即3s+d=5.7時，不知道d，所以無法判定，insufficient。 (3)由(1)(2)合併，s=1.65, d=0.75，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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143. If K is a positive three-digit integer, what is the hundreds digit of K? (1) The hundreds digit of K + 150 is 4. (2) The tens digit of K + 25 is 7. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: K的百位數字=? (1) 即K+150的百位數字=4時 400≦K+150≦499 => 250≦K≦349，可能為2或3 所以無法判定，insufficient。 (2) 即K+25的十位數字=7時 xxx + 25 = x7x， => x45≦K≦x54，不知道百位數 所以無法判定，insufficient。 (3) 由(1)(2)合併，K為250~254或345~349 還是無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------144. In the xy-plane, point P has coordinates (a, b) and point Q has coordinates (c, d). What is the distance between P and Q? (1) b - d = 4 (2) a - c = 3 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: PQ距離=? 即問(a-c)2+(b-d)2=? (1) 即b-d=4時，不知道a-c，所以無法判定，insufficient。 (2) 即a-c=3時，不知道b-d，所以無法判定，insufficient。 (3)由(1)(2)合併，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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145. If a, b, and c are positive integers, is b between a and c? (1) b is 3 greater than a, and b is 5 less than c. (2) c is 5 greater than b, and c is 8 greater than a. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: b在a和c之間? 直接在數線上討論最快 (1) 即b=a+3=c-5時 => a=b-3，c=b+5 => b在a, c之間，所以可以判定，sufficient。 (2) 即c=b+5=a+8時 => b=c-5，a=c-8 => b不在a, c之間，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------146. At a certain refreshment stand, all hot dogs have the same price and all sodas have the same price. What is the total price of 3 hot dogs and 2 sodas at the refreshment stand? (1) The total price of 5 sodas at the stand is less than the total price of 2 hot dogs. (2) The total price of 9 hot dogs and 6 sodas at the stand is \$21. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B 【思路】題目問: 3條熱狗和2杯汽水要多少錢? 問3h+2s=? (1)即5s<2h時，不知道價錢，所以無法判定，insufficient。 (2)即9h+6s=21時，=>3h+2s=7，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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147. If a certain company purchased computers at \$2,000 each and printers at \$300 each, how many computers did it purchase? (1) More than three printers were purchased. (2) The total amount for the purchase of the computers and the printers was \$15,000. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: 買了多少台電腦? 已知C*2,000+P*300，問C=? (1)即P>3時，但不知道電腦買了幾台，所以無法判定，insufficient。 (2)即C*2,000+P*300=15,000時，所以無法判定，insufficient。 (3)由(1)(2)合併，20C+3P=150，(20,3)|150，所以有整數解。 (C,P)=(3,30)或(6,10)，所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------148. If x = 3 and y = 6, is y > nx + k? (1) n = 5 (2) k = -10 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: y > nx + k ? 也就是問3n + k < 6? (1)即n=5時，不知道k，所以無法判定，insufficient。 (2)即k=-10時，不知道n，所以無法判定，insufficient。 (3)由(1)(2)合併，恆成立，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------149. Does x + c = y + c? (1) x = y (2) x = c (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】A 【思路】題目問: x + c = y + c? 也就是問x = y? (1)即x=y時，所以可以判定，sufficient。 (2)即x=c時，不知道y，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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150. If n is an integer between 3 and 9, what is the value of n? (1) On the number line, the distance from 3 to n is 2/3 of the distance from 3 to 9. (2) On the number line, n is 10 units to the right of -3 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: n=? 已知3≦n≦9 (1)即|n-3|= 4時，可知n=7或-1(不合)，所以可以判定，sufficient。 (2)即n=-3+10=7時，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------151. Is the average (arithmetic mean) of 5 different positive integers at least 30? (1) Each of the integers is a multiple of 10. (2) The sum of the 5 integers is 160. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: 5個不同正整數平均值至少30? (1) 即每一個都是10的倍數時 最小為10, 20, 30 ,40, 50，avg=30，所以可以判定，sufficient。 (2) 即avg=160/5=32時，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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152. 錯 A certain jar contains only b black marbles, w white marbles, and r red marbles. If one marble is to be chosen at random from the jar, is the probability that the marble chosen will be red greater than the probability that the marble chosen will be white? (1) r/(b+w) > w/(b+r) (2) b – w > r (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】A 【思路】題目問: 抽出red的機率大於抽出white的機率? 問r > w? (1) 即r/(b+w) > w/(b+r)時 可知r(b+r) >w(b+w) => rb+r2 > wb+w2 => b(r-w)+(r2- w2)＞0 => (r-w)(b+r+w) > 0 又b+r+w > 0，所以(r-w)>0，即 r > w 所以可以判定，sufficient。 (2) 即b – w > r時 不知道b，所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------153. On his trip from Alba to Benton, Julio drove the first x miles at an average rate of 50 miles per hour and the remaining distance at an average rate of 60 miles per hour. How long did it take Julio to drive the first x miles? (1) On this trip, Julio drove for a total of 10 hours and drove a total of 530 miles. (2) On this trip, it took Julio 4 more hours to drive the first x miles than to drive the remaining distance. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】A 【思路】題目問: 前x英哩花了多少時間? 已知前x miles速度為50(miles/h)，剩下y miles速度為60(miles/h)，問x/50=? (1) 即x/50+ y/60=10時 => 6x+5y=3,000又x+y=530 => x=350，所以可以判定，sufficient。 (2) 即x/50 - y/60=4時 => 6x-5y=1,200 不知道y，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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154. To install cable television in a home, a certain cable company charges a basic fee of \$30 plus a fee of \$20 for each cable outlet installed in the home. How much did the cable company charge the Horace family for installing cable television in their home? (1) The cable company installed three cable outlets in the Horace family home. (2) The amount that the cable company charged the Horace family for installing cable television in their home was equivalent to an average (arithmetic mean) charge of \$30 per cable outlet installed. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: 有線電視的安裝費? 即30+20n=? (1)即n=3時，所求=90，所以可以判定，sufficient。 (2)即30+20n=30*n時，n=3，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------155. If 500 is the multiple of 100 that is closest to x and 400 is the multiple of 100 that is closest to y, which multiple of 100 is closest to x + y? (1) x < 500 (2) y < 400 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: x+y最靠近哪個multiple of 100? multiple of 100 應該是 100, 200, 300, 400… 450<x<550，350<y<450 (1) 即x<500時 => 800< x+y <950 所以無法判定，insufficient。 (2) 即y<400時 => 800< x+y <950 所以無法判定，insufficient。 (3) 由(1)(2)合併 => 800< x+y <900 所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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156. Each of the numbers w, x, y, and z is equal to either 0 or 1. What is the value of w + x + y + z? (1) w/2 + x/4 + y/8 + z/16 = 11/16 (2) w/3 + x/9 + y/27 + z/81 = 31/81 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: w + x + y + z? 已知w, x, y, z不是0就是1 (1) 即8w+4x+2y+z=11時 (w, x, y, z)=(1, 0, 1, 1)為唯一解 所以可以判定，sufficient。 (2) 即27w+9x+3y+z=31時 (w, x, y, z)=(1, 0, 1, 1)為唯一解 所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------157. A certain list consists of five different integers. Is the average (arithmetic mean) of the two greatest integers in the list greater than 70? (1) The median of the integers in the list is 70. (2) The average of the integers in the list is 70. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: 最大兩數的平均值大於70? 已知5個數皆相異，直接代數字判定最快 (1) 即median=70時 因為相異所以恆成立，所以可以判定，sufficient。 (2) 即avg=70時 因為相異所以恆成立，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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158. A store purchased 20 coats that each cost an equal amount and then sold each of the 20 coats at an equal price. What was the store's gross profit on the 20 coats? (1) If the selling price per coat had been twice as much, the store's gross profit on the 20 coats would have been \$2,400. (2) If the selling price per coat had been \$2 more, the store's gross profit on the 20 coats would have been \$440. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B 【思路】題目問: 20隻羊的毛利潤? 已知Gross Profit=Selling Price - Purchasing Price 問20S - 20P=? (1)即20(2S) – 20P= 2,400時，所以無法判定，insufficient。 (2)即20(S+2) – 20P= 440時，=> 20S - 20P=400，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------159. Beth and Jim each received a salary increase. If Jim's salary was increased by the same percent as Beth's salary, did Beth receive a greater dollar increase in salary than Jim? (1) Before the increases, Jim’s salary was greater than \$25,000. (2) Before the increases, Jim’s salary was 4/5 of Beth’s salary. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B 【思路】題目問: Beth增加的薪水比Jim多? 已知(B2-B1)/B1= (J2-J1)/J1，問B2-B1> J2-J1? (1)即J1>25,000時，不知道B1和J1關係，所以無法判定，insufficient。 (2)即J1=(4/5)B1時，=> B1>J1，=> 分子 B2-B1> J2-J1，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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160. All the clients that Company X had at the beginning of last year remained with the company for the whole year. If Company X acquired new clients during the year, what was the ratio of the number of clients that Company X had at the end of last year to the number of clients that it had at the beginning of last year? (1) The ratio of the number of clients that Company X had at the beginning of last year to the number of new clients that it acquired during the year was 12 to 1. (2) Company X had 144 clients at the beginning of last year. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】A 【思路】題目問: 去年年末和年初的客戶比率是多少? 所求= (a+x) : a (1)即a:x=12:1時，所求=(12r+r) : 12r=13:12，所以可以判定，sufficient。 (2)即a=144時，不知道x，所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------161. 錯 In the xy-coordinate plane, line m and line k intersect at the point (4, 3). Is the product of their slopes negative? (1) The product of the x-intercepts of lines m and k is positive. (2) The product of the y-intercepts of lines m and k is negative. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: 斜率相乘為負? 已知直線相交於(4,3)，相交於第一象限，直接畫圖討論 (1) 兩直線的x截距乘積為正時，(+,0) (+,0)或(-,0) (-,0) 斜率可能為兩正，一正一負，兩負，所以無法判定，insufficient。 (2) 兩直線的y截距乘積為負時，(0,+) (0,-)或(0,-) (0,+) 斜率可能為兩正，一正一負，所以無法判定，insufficient。 (3) 由(1)(2)合併，斜率為一正一負，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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162. If n is a positive integer and r is the remainder when n2 - 1 is divided by 8, what is the value of r? (1) n is odd. (2) n is not divisible by 8. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】A 【思路】題目問: r=? 直接代數字進去判斷最快 (1) 即n=2k+1時 可知n2 – 1 = 4k(k+1)，又k(k+1)為2的倍數 r=0，所以可以判定，sufficient。 (2) n=1時，r=0 n=2時，r=3 n=3時，r=0 n=4時，r=7 所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------163. In a certain election, 240 men and 280 women voted for the winning candidate. What was the total number of men and women who voted in the election? (1) The number of women who voted was 7/8 the number of men who voted. (2) Of the men and women who voted, 30 percent of the men and 40 percent of the women voted for the winning candidate. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B 【思路】題目問: 選舉中總投票人數? 已知WM=240, WF=280，問WM+WF+LM+LF=? (1)即WF+LF=(7/8)(WM+LM)時 =>280+LF=(7/8)(240+LM)，所以無法判定，insufficient。 (2)即0.3*(WM+LM)=240, 0.4*(WF+LF)=280時 => WM+LM=800, WF+LF=700，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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164. If mv < pv < 0, is v > 0? (1) m < p (2) m < 0 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: v > 0? 已知 mv < pv < 0 即 mv < 0，pv < 0 (1) 即m < p時 =>可知 v > 0，不然不等式方向會改變 所以可以判定，sufficient。 (2) 即m < 0時 又 mv < pv < 0 => v > 0， 所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------165. If x and y are points on the number line, what is the value of x + y? (1) 6 is halfway between x and y. (2) y = 2x (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】A 【思路】題目問: x + y=? (1) 即6為x, y中點時 可知x+y=12，所以可以判定，sufficient。 (2) 即y=2x時，不知道x，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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166. 錯 If the prime numbers p and t are the only prime factors of the integer m, is m a multiple of p2t? (1) m has more than 9 positive factors. (2) m is a multiple of p3. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B 【思路】題目問: m是p2 t的倍數? 已知m=pa * tb，a≧1，b≧1，又若m是p2 t的倍數時，則a≧2。 (1) 即(a+1)(b+1)>9時 無法判斷a≧2 所以無法判定，insufficient。 (2) 即m是p3的倍數時，a≧3，原式恆成立 所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------167. A certain circular area has its center at point P and has radius 4, and points X and Y lie in the same plane as the circular area. Does point Y lie outside the circular area? (1) The distance between point P and point X is 4.5. (2) The distance between point X and point Y is 9. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: Y點在圓外? (1) 即P點到X點距離4.5時 但不知道Y的關係 所以無法判定，insufficient。 (2) 即X點到Y點距離9時 但不知道X和Y的關係 所以無法判定，insufficient。 (3) 由(1)(2)合併，無論如何Y都會在圓外 所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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168. In a certain senior class, 72 percent of the male students and 80 percent of the female students have applied to college. What fraction of the students in the senior class is male? (1) There are 840 students in the senior class. (2) 75 percent of the students in the senior class have applied to college. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B 【思路】題目問: M/M+F的比例? (1) 即M+F=840時 不知道M，所以無法判定，insufficient。 (2) 即0.72*M+0.8*F=0.75*(M+F)時 則0.05F=0.03M，F:M=3:5 所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------169. Greta and Randy collected bottles to be recycled. How many bottles did Randy collect? (1) Greta and Randy collected a total of 85 bottles. (2) Greta collected 15 more bottles than Randy did. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: Randy收集多少瓶子? 問R=? (1)即G+R=85時，不知道G，所以無法判定，insufficient。 (2)即G=R+15時，不知道G，所以無法判定，insufficient。 (3)由(1)(2)合併，G=50, R=35，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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170. If zt < -3, is z < 4? (1) z < 9 (2) t < -4 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: z < 4? 已知 zt < -3 (1)即z < 9時，(z, t)=(+,-)或(-,+)，但不知道t，所以無法判定，insufficient。 (2)即t < -4時，只知z>0，所以無法判定，insufficient。 (3)由(1)(2)合併，只知 0 < z < 9，所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------171. If k is a positive integer and the tens digit of k + 5 is 4, what is the tens digit of k? (1) k > 35 (2) The units digit of k is greater than 5. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B 【思路】題目問: k的十位數字是多少? 已知40≦k+5≦49 => 35≦k≦44 (1)即k>35時，所求=3或4，所以無法判定，insufficient。 (2)即k的個位數字大於5時，可知36≦k≦39，所求=3，可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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172. If a and b are nonzero numbers on the number line, is 0 between a and b? (1) The distance between 0 and a is greater than the distance between 0 and b. (2) The sum of the distances between 0 and a and between 0 and b is greater than the distance between 0 and the sum a + b. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B 【思路】題目問: 0在a, b之間? 直接在數線上畫圖討論 (1)即|a-0|>|b-0|時，所以無法判定，insufficient。 (2)即|a-0|+|b-0|>|a+b-0|時 =>|a|+|b|>|a+b| => a, b必為一正一負才能成立，不然等號會成立 所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------173. In the xy-plane, line k passes through the point (1,1) and line m passes through the point (1,-1). Are lines k and m perpendicular to each other? (1) Lines k and m intersect at the point (1,-1). (2) Line k intersects the x-axis at the point (1,0). (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: 直線k和直線m互相垂直? 已知直線k通過(1,1)，直線m通過(1,-1) (1) 即直線k和直線m相交於(1,-1)時 只知道直線k方向，但不知道直線m方向，所以無法判定，insufficient。 (2) 即直線k和x軸相交於(1,0)時 只知道直線k方向，但不知道直線m方向，所以無法判定，insufficient。 (3)由(1)(2)合併，還是不知道直線m方向，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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174. Each week Connie receives a base salary of \$500, plus a 20 percent commission on the total amount of her sales that week in excess of \$1,500. What was the total amount of Connie's sales last week? (1) Last week Connie's base salary and commission totaled \$1,200. (2) Last week Connie's commission was \$700. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: 上禮拜的銷售額? 已知500+(x-1,500)*20%，問x=? (1) 即500+(x-1,500)*20%=1,200時 => x=5,000，所以可以判定，sufficient。 (2) 即(x-1,500)*20%=700時 => x=5,000，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------175. Ann bought five different kinds of fruit: apples, oranges, pears, mangoes, and bananas. If the number of apples that Ann bought was twice the number of oranges and if the number of pears that Ann bought was the same as the number of apples and oranges combined, what fraction of the total number of pieces of fruit that Ann bought were pears? (1) Ann bought a total of 18 pieces of fruit. (2) Ann bought 5 bananas. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: 梨子在五種水果中所佔的比率? 已知A=2O，P=A+O，問P/ A+O+P+M+B=? (1)即A+O+P+M+B=18時，不知道P, A, O，所以無法判定，insufficient。 (2)即B=5時，但不知道其他資訊，所以無法判定，insufficient。 (3)由(1)(2)合併，還是資訊不足，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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176. In 1995 Division A of Company X had 4,850 customers. If there were 86 service errors in Division A that year, what was the service-error rate, in number of service errors per 100 customers, for Division B of Company X in 1995? (1) In 1995 the overall service-error rate for Divisions A and B combined was 1.5 service errors per 100 customers. (2) In 1995 Division B had 9,350 customers, none of whom were customers of Division A. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: B部門的錯誤率? 已知CA=4,850，EA=86，問EB/CB=? (1)即(86+EB)/(4,850+CB)=1.5/100時，但不知道EB和CB，所以無法判定，insufficient。 (2)即CB=9,350時，但不知道EB，所以無法判定，insufficient。 (3)由(1)(2)合併，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------177. If x and y are positive integers, is x an even integer? (1) x (y + 5) is an even integer. (2) 6y2+ 41y + 25 is an even integer. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: x是偶數? (1) 即x (y + 5)是偶數時 可能x為偶數或y+5為偶數，所以無法判定，insufficient。 (2) 即6y2+41y+25是偶數時 =>41y為奇數 => y為奇數 但不知道x，所以無法判定，insufficient。 (3) 由(1)(2)合併，只知y為奇數，不知道x，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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178.

In the figure shown, point O is the center of the semicircle and points B, C, and D lie on the semicircle. If the length of line segment AB is equal to the length of line segment OC, what is the degree measure of angle BAO? (1) The degree measure of angle COD is 60. (2) The degree measure of angle BCO is 40. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: 角BAO=? 已知AB = OC = OB，角BAO =角BOA = x 角ABO = 180 – 2x， 因此角CBO = 2x. 因為OC = OB，角CBO =角BCO =2x，角BOC = 180 – 4x 角COD = 180 - (180-4x) - x = 3x (1)即3x=60時，可知x=20，所以可以判定，sufficient。 (2)即2x=40時，可知x=20，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------179. Working independently at their respective constant rates, pumps X and Y took 48 minutes to fill an empty tank with water. What fraction of the water in the full tank came from pump X? (1) Working alone at its constant rate, pump X would have taken 80 minutes to fill the tank with water. (2) Working alone at its constant rate, pump Y would have taken 120 minutes to fill the tank with water. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: 水槽的水有多少是來自於x幫浦? 已知(1/tx)+(1/ty)=1/48，所求=Vx / (Vx+Vy)=? (1) 即tx=80時，=>ty=120 所求=1/80 / (1/80 + 1/120)=3/5，所以可以判定，sufficient。 (2) 即ty=120時，=>tx=80 所求=1/80 / (1/80 + 1/120)=3/5，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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180. What is the value of v3 - k3? (1) vk > 0 (2) v - k = 6 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: v3 - k3=? 已知v3 - k3=(v-k)3- 3vk(v-k) (1) 即vk > 0時，不知道v-k，所以無法判定，insufficient。 (2) 即v - k = 6時，不知道vk，所以無法判定，insufficient。 (3) 由(1)(2)合併，不知道vk，還是無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------181. Rasheed bought two kinds of candy bars, chocolate and toffee, that came in packages of 2 bars each. He handed out 2/3 of the chocolate bars and 3/5 of the toffee bars. How many packages of chocolate bars did Rasheed buy? (1) Rasheed bought 1 fewer package of chocolate bars than toffee bars. (2) Rasheed handed out the same number of each kind of candy bar. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: 買了多少包巧克力棒? 已知發出(2/3)(2C)和(3/5)(2T)，問C=? 注意，看清楚題目單位敘述。 (1)即C=T-1時，不知道T，所以無法判定，insufficient。 (2)即(2/3)(2C)=(3/5)(2T)時，=> C:T=9:10，所以無法判定，insufficient。 (3)由(1)(2)合併，可知C=9，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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182. If m and r are two numbers on a number line, what is the value of r? (1) The distance between r and 0 is 3 times the distance between m and 0. (2) 12 is halfway between m and r. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: r=? (1)即|r-0|=3|m-0|時，可知道|r|=3|m|，r=3m或-3m，所以無法判定，insufficient。 (2)即m+r=24時，不知道m，所以無法判定，insufficient。 (3)由(1)(2)合併， => (m,r)=(6, 18)或(-12, 36)，所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------183. If m is a positive odd integer between 2 and 30, then m is divisible by how many different positive prime numbers? (1) m is not divisible by 3. (2) m is not divisible by 5. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】A 【思路】題目問: m有多少個正質因數? 已知 2 < m < 30，可以直接代入數字判斷 (1) 即m不被3整除時 m可以為 5, 7, 11, 13, 17, 19, 23, 25, 29。 所有這些數都是質數，只有一個質因數 所以可以判定，sufficient。 (2) 即m不被5整除時 m可以為 3, 7, 9, 11, 13, 17, 19, 21, 23, 27, 29。 m有不同的質因數個數 所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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184. 錯 If k is an integer greater than 1, is k equal to 2r for some positive integer r? (1) k is divisible by 26. (2) k is not divisible by any odd integer greater than 1. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B 【思路】題目問: k = 2r? 已知k > 1 (1) 即k=26*m時 不知道m是否等於2 所以無法判定，insufficient。 (2) 即k不能被任何大於1的奇數整除時 => 只知k只可能被偶數整除 => 2, 4, 6(不合), 8, 10(不合),12(不合), 14(不合), 16, …整除， 看出其規律性，k只能為 21, 22, 23, 24, … 所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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185. 錯 Of the 200 members of a certain association, each member who speaks German also speaks English, and 70 of the members speak only Spanish. If no member speaks all three languages, how many of the members speak two of the three languages? (1) 60 of the members speak only English. (2) 20 of the members do not speak any of the three languages. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: 多少人會講兩種語言? 已知G=0，因為他們都會說英語 S=70，題目所給 (G∩E∩S)=0，題目所給 又會講German的人也會講English => (G∩S)= 0 問(G∩E)+ (E∩S)+ (G∩S)- 3(G∩E∩S) =(G∩E)+ (E∩S)=?

(1)即E=60時 所求= (G∩E)+ (E∩S) 不知道(G∩E) 和(E∩S)，所以無法判定，insufficient。 (2)即(G’∩E’∩S’) = (G∪E∪S)’ = 全 - (G∪E∪S) = 200 - (G∪E∪S) =20時 (G∪E∪S)= G + E + S - (G∩E) - (E∩S) - (G∩S) + (G∩E∩S) = 180  0 + E + 70 - (G∩E) - (E∩S)- 0 + 0 = 180  E - (G∩E) - (E∩S) = 110 所求= (G∩E)+ (E∩S)= 110 – E 不知道E，所以無法判定，insufficient。 (3) 由(1)(2)合併，E=60，所求= 110 – E = 50，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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186. If a, b, and c are integers, what is the value of a? (1) (a - 7)(b - 7)(c - 7) = 0 (2) bc = 18 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: a=? 已知a, b, c為整數，問a=? (1) 即(a-7)(b-7)(c-7) = 0時 可能a=7, or b=7, or c=7，所以無法判定，insufficient。 (2) 即bc = 18時，不知道a，所以無法判定，insufficient。 (3) 由(1)(2)合併，可知b和c不能為7 => a=7，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------187. An antique dealer bought a coffee table that was then sold for a profit. What was the selling price of the coffee table? (1) The dealer's cost for the coffee table was \$340. (2) The dealer's gross profit on the coffee table was 15% of the selling price. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: 咖啡桌的售價? 已知Selling Price – Cost = Profit，問S=? (1)即C=340時，但不知道P，所以無法判定，insufficient。 (2)即P=0.15*S時，但不知道C，所以無法判定，insufficient。 (3)由(1)(2)合併，S - 340=0.15S，S=400，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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188. Is x4 + y4 > z4? (1) x2 + y2 > z2 (2) x + y > z (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: x4 + y4 > z4? 心得: DS的作題技巧 Step 1: 只要代入數字檢驗時，就是要找出 insufficient的情形。 Step 2: 如果statement是sufficient的情形，詴詴看能不能用幾何或其它定理來證明成立。 (1) 即x2 + y2 > z2時 兩邊同時平方，=> (x2 + y2)2> (z2)2 => x4 +2 x2y2 + y4 > z4， 因為2 x2y2> 0，所以無法判定，insufficient。 (2) 即x + y > z時 兩邊同時四次方，=> (x+ y)4> z4 => x4 +4 x3y+6 x2y2+4 xy3+y4> z4， 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 因為4 x3y+6 x2y2+4 xy3 = 2x2y2 [2(x/y)+3+2(y/x) ] 又 [ t+(1/t) / 2] ≧ root[ t*(1/t)] => t+(1/t) ≧ 2 所以4 x3y+6 x2y2+4 xy3> 0，所以無法判定，insufficient。 (3)由(1)(2)合併，還是無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------189. In the xy-plane, the line k passes through the origin and through the point (a, b), where ab ≠ 0. Is b positive? (1) The slope of line k is negative. (2) a < b (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: b>0? 已知ab≠ 0，斜率=(b-0)/(a-0)=b/a (1) 即(b/a)<0時 表示(a,b)=(+,-)或(-,+) 所以無法判定，insufficient。 (2) 即a<b時 不知道b的正負，所以無法判定，insufficient。 (3) 由(1)(2)合併，(a,b)=(-,+)，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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190. The numbers of books read by 7 students last year were 10, 5, p, q, r, 29, and 20. What was the range of the numbers of books read by the 7 students last year? (1) 5 < p < q (2) p < r < 15 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: 7個學生所看的書其值域是? 已知值域 = 最大值 - 最小值 (1) 即5<p<q時 不知道r，無法確認最大值和最小值 所以無法判定，insufficient。 (2) 即p<r<15時 不知道q，無法確認最大值和最小值 所以無法判定，insufficient。 (3) 由(1)(2)合併，最小值=5，最大值可能為29或q，所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------191. Is the positive integer j divisible by a greater number of different prime numbers than the positive integer k? (1) j is divisible by 30. (2) k = 1,000 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: j的質因數會比k的質因數多? (1)即j=30n=2*3*5*n時，不知道k，所以無法判定，insufficient。 (2)即k=1,000=23*53時，不知道j，所以無法判定，insufficient。 (3)由(1)(2)合併，j至少有3個質因數，k有2個質因數，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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192. Is xy > 0? (1) x - y > -2 (2) x - 2y < -6 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: xy>0? (1) 即x-y > -2時，(x,y)可以為(+,+)(+,-)(-,-)，所以無法判定，insufficient。 (2) 即x-2y < -6時，(x,y)可以為(+,+)(-,+)(-,-)，所以無法判定，insufficient。 (3) 由(1)(2)合併，(x,y)可以為(+,+)(-,-)，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------193. If x - y > 10, is x - y > x + y ? (1) x = 8 (2) y = -20 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: x - y > x + y? 已知x - y > 10，問 -y > y? (1) 即x = 8時 可知 y < -2，恆成立 所以可以判定，sufficient。 (2) 即y = -20時 恆成立，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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194. If r is the remainder when the positive integer n is divided by 7, what is the value of r? (1) When n is divided by 21, the remainder is an odd number. (2) When n is divided by 28, the remainder is 3. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B 【思路】題目問: n除以7的餘數? (1) 即n=21a+k，其中k是奇數時 不知道k，所以無法判定，insufficient。 (2) 即n=28b+3=7(4b)+3 => r=3，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------195. 錯 If w, x, y, and z are integers such that w/x and y/z are integers, is w/x + y/z odd? (1) wx + yz is odd. (2) wz + xy is odd. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B 【思路】題目問: (w/x) + (y/z)是奇數? 即(w/x) + (y/z) = (wz+xy) / xz 是奇數? (1) 即wx + yz 為奇時 (wx, yz)=(奇,偶)或(偶,奇) (i) (wx,yz)=(奇,偶) (w,x)= (奇,奇)，(y,z)= (奇,偶)(偶,奇)(偶,偶) 因為 w/x , y/z 為整數，(w,x)= (奇,奇)，(y,z)= (偶,奇)或(偶,偶) => (w/x)為奇, (y/z)為偶或奇 所求=奇或偶，所以無法判定，insufficient。 (ii) (wx,yz)= (偶,奇)， 同上(w,x)= (奇,偶)或(偶,奇)或(偶,偶)，(y,z)= (奇,奇) 因為 w/x , y/z 為整數，(w,x)= (偶,偶)，(y,z) =(奇,奇) => (w/x)為奇或偶, (y/z)為奇 所求=偶或奇，所以無法判定，insufficient。 所以無法判定，insufficient。 (2) 即wz + xy 為奇時 又w/x , y/z 為整數且所求為一整數解 => xz 必為奇數 所求=奇，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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196. A recent lunch meeting at a certain club was attended by members and guests. Each member paid \$4 for the lunch, and each guest paid \$8 for the lunch. How many of the people attending the meeting were members? (1) A total of 20 people attended the meeting. (2) A total of \$92 was paid for the lunch. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: 多少與會的人是俱樂部成員? 已知4M+8G，問M=? (1)即M+G=20時，不知道G，所以無法判定，insufficient。 (2)即4M+8G=92時，不知道G，所以無法判定，insufficient。 (3)由(1)(2)合併，G=3，M=17，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------197. 錯 The integers m and p are such that 2 < m < p and m is not a factor of p. If r is the remainder when p is divided by m, is r > 1? (1) The greatest common factor of m and p is 2. (2) The least common multiple of m and p is 30. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】A 【思路】題目問: 餘數 > 1? 已知2 < m < p，m不是p的因數，p=mq+r => r 不為0 (1) 即(m,p)=2時 又m=2h, p=2k, (h,k)=1, p=mq+r => 2k=2hq+r => r=2(k-hq) => r是2的倍數 所以可以判定，sufficient。 (2) 即[m,p]=30=2*3*5=dhk時 令m=3, p=10 => r=1 m=6, p=15 => r=3 所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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198. If @ denotes one of the four arithmetic operations addition, subtraction, multiplication and division, what is the value of 1 @ 2? (1) n @ 0 = n for all integers n. (2) n @ n = 0 for all integers n. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】B 【思路】題目問: 1 @ 2=? (1) 即n @ 0 = n時，@可以為+或 -，所以無法判定，insufficient。 (2) 即n @ n = 0時，@只能為-，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------199. 錯 Warehouse W's revenue from the sale of sofas was what percent greater this year than it was last year? (1) Warehouse W sold 10 percent more sofas this year than it did last year. (2) Warehouse W's selling price per sofa was \$30 greater this year than it was last year. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: 今年收入比去年成長多少? 今年收入=n1*s1，去年收入=n0*s0，所求= (n1*s1- n0*s0)/ (n0*s0) (1) 即n1=1.1n0時， 所求=(1.1n0*s1- n0*s0)/ (n0*s0)=(1.1s1-s0)/s0 不知道s1，所以無法判定，insufficient。 (3) 即s1=s0+30時， 所求=[n1*(s0+30)- n0*s0]/ (n0*s0) = [n1s0+30n1 - n0*s0]/ (n0*s0)= [(n1 - n0)*s0+30n1]/ (n0*s0) 不知道n1，所以無法判定，insufficient。 (3)由(1)(2)合併，所求=(1.1s1-s0)/s0= [1.1(s0+30)-s0]/s0=[0.1s0+33]/s0 不知道s0，還是無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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200. 錯 If d is a positive integer and f is the product of the first 30 positive integers, what is the value of d? (1) 10d is a factor of f. (2) d > 6 (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: d的值? 已知f = 30! (1) 即10d是f的因數時 30!=2x * 5y=226 * 57 d可以為1, 2, …, 7 所以無法判定，insufficient。 (2) 即d可以為任何大於6的數字時 所以無法判定，insufficient。 (3) 由(1)(2)合併，d=7，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------201. If k is a line in the xy-plane, what is the slope of k? (1) The x-intercept of k is 2. (2) The y-intercept of k is 3. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: 直線k斜率=? 已知y=ax+b，問a=? (1)即x截距為2時，(2, 0)代入=>2a+b=0，所以無法判定，insufficient。 (2)即y截距為3時，(0, 3)代入=>b=3，所以無法判定，insufficient。 (3)由(1)(2)合併，a=-3/2，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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202. The numbers x and y are not integers.

The value of x is closest to which integer?

(1) 4 is the integer that is closest to x + y. (2) 1 is the integer that is closest to x - y. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: x靠近哪個整數? (1) 即4是最靠近x+y的整數時 => 3.5 < x+y < 4.5，所以無法判定，insufficient。 (2) 即1是最靠近x-y的整數時 => 0.5 < x-y < 1.5，所以無法判定，insufficient。 (3) 由(1)(2)相加，4 < 2x < 6，=> 2 < x < 3，還是無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------203. What is the result when x is rounded to the nearest hundredth? (1) When x is rounded to the nearest thousandth the result is 0.455. (2) The thousandths digit of x is 5. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問:四捨五入到最接近的百分位數的x是多少? (1)即四捨五入到最接近的千分位數的x=0.455時 => 0.4545 ≦ x < 0.4555 所求可能為0.45或0.46，所以無法判定，insufficient。 (2) 即x的千分位數為5時 只知道四捨五入後會進1，但還是不知道x值，所以無法判定，insufficient。 (3)由(1)(2)合併，x=0.455…，所求=0.46，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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204. If Mary always takes the same route to work, how long did it take Mary to get to work on Friday? (1) It took Mary 20 minutes to get to work on Thursday. (2) Mary's average speed on her trip to work was 25 percent greater on Thursday than it was on Friday. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: Mary禮拜五花了多久時間到工作地點? 問t5=? (1) 即禮拜四花了20分鐘到工作地點時 不知道禮拜五的狀況，所以無法判定，insufficient。 (2) 即禮拜四平均速度比禮拜五快了25%時 不知道時間的關係，所以無法判定，insufficient。 (3) 由(1)(2)合併，V4 : V5=5:4，=> t4 : t5 = 4:5 20 : t5 = 4:5，=> t5=25，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------205. What was a certain company's revenue last year? (1) Last year the company's gross profit was \$4,100. (2) Last year the company's revenue was 50 percent greater than its expenses. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: 去年公司的收入? 已知Revenue – Cost = Gross Profit，問R=? (1)即R - C=4,100時，所以無法判定，insufficient。 (2)即R =1.5C時，所以無法判定，insufficient。 (3)由(1)(2)合併，C=8,200，=> R=12,300，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

98

206. In 1984 a certain union had a total of 15,600 members. Was the percent increase in the total number of members in the union from 1984 to 1985 greater than that from 1985 to 1986? (1) From 1984 to 1985 the total number of members in the union increased by 781, and from 1985 to 1986 the total number of members in the union again increased by 781. (2) In 1985 the union had a total of 16,381 members, and in 1986 the union had a total of 17,162 members. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: 1984 to 1985人口增加的比率大於1985 to 1986? 已知f(1984)=15,600，問[f(1985)-f(1984)]/ f(1984) > [f(1986)-f(1985)]/ f(1985)? (1)即[f(1985)-f(1984)]= [f(1986)-f(1985)]=781時 => f(1985)= 15,600+781，f(1986)= f(1985)+781= 15,600+781+781 所以可以判定，sufficient。 (2)即f(1985)=16,381，f(1986)=17,162時 所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------207. At a certain theater, the cost of each adult's ticket is \$5 and the cost of each child's ticket is \$2. What was the average (arithmetic mean) cost of all the adults' and children's tickets sold at the theater yesterday? (1) Yesterday the ratio of the number of children's tickets sold at the theater to the number of adults' tickets sold at the theater was 3 to 2. (2) Yesterday 80 adults' tickets were sold at the theater. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】A 【思路】題目問: 昨晚的大人和小孩的平均花費? 即(x*5+y*2)/(x+y)=? (1)即x:y=3:2時，所求=19/5，所以可以判定，sufficient。 (2)即x=80時，不知道y，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

99

208. At a certain store, each notepad costs x dollars and each marker costs y dollars. markers, is \$10 enough to buy 4 notepads and 4 markers instead?

(1) Each notepad costs less than \$1. (2) \$10 is enough to buy 11 notepads. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: 10元可以買4本筆記本和4個書籤? 已知5x+3y<10，問4x+4y<10? (1) 即x < 1時，=> 5x < 5 => 3y < 5，=> y < 5/3 => 4x+4y < 4+ 20/3 = 32/3 = 10又1/3 所以無法判定，insufficient。 (2) 即11x < 10時，=> x < 10/11，=> 5x < 50/11 => 3y < 10 - 50/11 = 60/11，=> y < 20/11 => 4x+4y < 40/11+80/11 = 120/11 = 10又10/11 所以無法判定，insufficient。 (3)由(1)(2)合併，x < 10/11，y < 20/11，=> 還是無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------209. The cost of each adult's ticket for a certain concert was \$30, and the cost of each child's ticket for the concert was \$24. If Hannah purchased tickets for this concert, what was the average (arithmetic mean) cost per ticket? (1) Hannah purchased twice as many children's tickets as adults' tickets. (2) Hannah purchased 4 children's tickets. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】A 【思路】題目問: 每張票的算術平均? 已知(30x+24y)/(x+y)=? (1)即x=2y時，所以可以判定，sufficient。 (2)即y=4時，不知道x，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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210. For each order, a mail-order bookseller charges a fixed processing fee and an additional shipping fee for each book in the order. Rajeev placed five different orders with this bookseller--an order for 1 book in January, an order for 2 books in February, an order for 3 books in March, an order for 4 books in April, and an order for 5 books in May. What was the total of Rajeev's processing and shipping fees for these five orders? (1) Rajeev's processing and shipping fees were \$1.00 more for his order in March than for his order in January. (2) The total of Rajeev's shipping fees for the five orders was \$7.50. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: 五筆訂單的手續費和運費? 已知: Jan => 1 book: x+y Feb => 2 books: x+2y Mar => 3 books: x+3y Apr => 4 books: x+4y May => 5 books: x+5y 問5x+15y=? (1)即(x+3y) – (x+y) =1時，=> y=0.5，不知道x，所以無法判定，insufficient。 (2)即15y=7.5時，=> y=0.5，y=0.5，不知道x，所以無法判定，insufficient。 (3)由(1)(2)合併，還是無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------211. A certain economics class consists of 50 women and 30 men. How many of the men in the class are business majors? (1) 40 percent of the women in the class are business majors. (2) 50 percent of all the people in the class are business majors. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: 多少男生主修企管? (1) 即主修企管的女生有50*40%=20人時 不知道男生的情形，所以無法判定，insufficient。 (2) 即主修企管有80*50%=40人時 不知道男生的情形，所以無法判定，insufficient。 (3) 由(1)(2)合併，所求=40-20=20，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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212. For a certain play performance, adults' tickets were sold for \$12 each and children's tickets were sold for \$8 each. How many children's tickets were sold for the performance? (1) The total revenue from the sale of adults’ and children’s tickets for the performance was \$5,040. (2) The number of adults’ tickets sold for the performance was 1/3 the total number of adults’ and children’s tickets sold for the performance. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: 表演賣了多少張兒童票? 問C=? (1)即12A+8C=5,040時，不知道A，所以無法判定C，insufficient。 (2)即A=1/3(A+C)時，=>2A=C，所以無法判定，insufficient。 (3)由(1)(2)合併，C=360，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------213. Linda, Robert, and Pat packed a certain number of boxes with books. What is the ratio of the number of boxes of books that Robert packed to the number of boxes of books that Pat packed? (1) Linda packed 30 percent of the total number of boxes of books. (2) Robert packed 10 more boxes of books than Pat did. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: R:P=? (1)即L=0.3(R+P+L)時，所以無法判定，insufficient。 (2)即R-P=10時，所以無法判定，insufficient。 (3)由(1)(2)合併，還是無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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214. 錯 Is the measure of one of the interior angles of quadrilateral ABCD equal to 60 degrees? (1) Two of the interior angles of ABCD are right angles. (2) The degree measure of angle ABC is twice the degree measure of angle BCD. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問:四邊形有一個內角為60度? 需要知道一角是否等於60度，就要知道其他三角，或是相對關係。 (1) 即有兩個角是直角時 只能知道另外兩角相加為180度 所以無法判定，insufficient。 (2) 即某一角是隔壁角的兩倍時 這種關係的角度組合方式有很多種，如(60, 30)，(90, 45)或(60, 120)， 所以無法判定，insufficient。 (3) 由(1)(2)合併，角度可能為 90°, 90°, x°, 2x° 第一種是(2)發生在未知的兩角，此時另外兩角為120與60，此時四個角是90, 90, 60, 120； 第二種是(2)發生在一角為90度的情況下，那另一角即為45度，此時四個角是90, 90, 45, 135； 不一定有內角等於60度，所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------215. 錯 On the number line, the distance between point A and point C is 5 and the distance between point B and point C is 20. Does point C lie between point A and point B? (1) The distance between point A and point B is 25. (2) Point A lies to the left of point B. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】A 【思路】題目問: C點在AB之間? 已知|A-C|=5，|B-C|=20 直接畫出數線來判斷 A 5 C 20 B (i) ---------|--------|----------------|----------C 5 A 15 B (ii) ---------|--------|----------------|----------(1) 即|A-B|=25時 C必在AB之間，所以可以判定，sufficient。 (2) 即A在B點左方時 有兩種情形，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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216. 錯 Is the three-digit number n less than 550? (1) The product of the digits in n is 30. (2) The sum of the digits in n is 10. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】C 【思路】題目問: 三位數n < 550? 已知n=x*100+y*10+z，1≦x, y, z≦9 (1) 即xyz=30=2*3*5時 xyz=2*3*5或1*5*6 所以無法判定，insufficient。 (2) 即x+y+z=10時， 所以無法判定，insufficient。 (3) 由(1)(2)合併，xyz=2*3*5，n={235, 253, 325, 352, 523, 532} 所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------217. If n is a three-digit positive integer, what is the sum of the digits of n? (1) The hundreds digit of n is 3 times the units digit. (2) The hundreds digit of n is 3 more than the tens digit. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】E 【思路】題目問: 三個數n的位數值總和? 已知n=x*100+y*10+z，1≦x, y, z≦9，問x+y+z=? (1)即x=3z時，不知道y，所以無法判定，insufficient。 (2)即x=y+3時，不知道z，所以無法判定，insufficient。 (3)由(1)(2)合併，3z=y+3時，(x,y,z)=(3,2,1)(6,3,2)(9,6,3)，還是無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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218. The retail price of a certain refrigerator is 1.6 times its wholesale price. What is the difference between the retail price and the wholesale price of the refrigerator? (1) The wholesale price of the refrigerator is \$200. (2) The retail price of the refrigerator is \$320. (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are not sufficient. -------------------------------------------------------------------------------------------------------------------------------------------------------【答案】D 【思路】題目問: 冰箱的批發價=? 已知R=1.6W，問R-W=? (1)即W=200時，=> R=320，所以可以判定，sufficient。 (2)即R=320時，=> W=200，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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