Practice Test #1 Data Sufficiency (218 Questions) 1. A garden store purchased a number of shovels and a number of rakes. If the cost of each shovel was \$14 and the cost of each rake was \$9, what was the total cost of the shovels and rakes purchased by the store? (1) The ratio of the number of shovels to the number of rakes purchased by the store was 2 to 3. (2) The total number of shovels and rakes purchased by the store was 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】C 【思路】題目問: 14s + 9r =? (1) 即s:r=2:3時 只知比例，無法求得數量 所以無法判定，insufficient。 (2) 即s+r=50時 只知總和，不知數量各為多少 所以無法判定，insufficient。 (3) 由(1)(2)合併，可得a=10，即s=20，r=30，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------2. Of the students who eat in a certain cafeteria, each student either likes or dislikes lima beans and each student either likes or dislikes brussels sprouts. Of these students, 2/3 dislike lima beans; and of those who dislike lima beans, 3/5 also dislike brussels sprouts. How many of the students like brussels sprouts but dislike lima beans? (1) 120 students eat in the cafeteria. (2) 40 of the students like lima beans. (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: (2/3)s *(1 - 3/5) =?

sprouts ~ sprouts

beans a c

~ beans b d (2/3)s

(1) 即s=120時 所以可以判定，sufficient。 (2) 即s*(1 - 2/3)=40時 => s=120，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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3. How many different prime numbers are factors of the positive integer n? (1) Four different prime numbers are factors of 2n. (2) Four different prime numbers are factors of n2. (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的不同質因數個數=? (1) 2n有四個不同質因數時 即 2n = 2x * 3y * 5z * 7w (i) 2n= 2 * 3 * 5 * 7 時 n = 3 * 5 * 7 => n的不同質因數個數=3。 (ii) 2n=22 * 3 * 5 * 7 時 n= 2 * 3 * 5 * 7 => n的不同質因數個數=4。 所以無法判定，insufficient。 (2) n2有四個不同質因數時 即n2 = 2x * 3y * 5z * 7w n平方之後，所有的質因數也會跟著平方 => 質因數個數並不會變多或變少。 所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------4. Is the product of a certain pair of integers even? (1) The sum of the integers is odd. (2) One of the integers is even and the other 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 *y =偶數? (1) 即總和為奇時 表示必為一奇一偶，所以可以判定，sufficient。 (2) 即一奇一偶時 所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

2

5. If k is an integer and 2 < k < 8, what is the value of k? (1) k is a factor of 30. (2) k is a factor of 12. (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值? (1) 即k是30的因數時 k=3, 5, 6，所以無法判定，insufficient。 (2) 即k是12的因數時 k=3, 4, 6，所以無法判定，insufficient。 (3)由(1)(2)合併，可得k=3, 6，所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------6.

On the number line above, is the product of w, x, y, and z negative? (1) z is positive. (2) The product of w and x 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: w*x*y*z*為負? (1)不知w, x, y的正負，所以無法判定，insufficient。 (2) (w, x) =(+,+) or (-,-)，不知 y, z的正負，所以無法判定，insufficient。 (3)由(1)(2)合併，y可能為正或負，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

3

7. If y and z are integers, is y(z + 1) odd? (1) y is odd (2) 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: y(z+1)為奇數? (1) 即y為奇時 z+1無法確定奇偶，所以無法判定，insufficient。 (2) 即z為偶時 z+1為奇，但y無法確定奇偶，所以無法判定，insufficient。 (3)由(1)(2)合併，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------8. If n and m are positive integers, what is the remainder when 3 4n + 2 + m is divided by 10? (1) n = 2 (2) m = 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: 34n + 2 + m 除以10的餘數? 31=3，32=9，33=27，34=81，35=81 => 餘數為3, 9, 7, 1，每四項一個循環 (1) n=2時，310+ m除以10的餘數可能為7, 1, 9, 3 所以無法判定，insufficient。 (2) m=1時，33+ 4n除以10的餘數只能為7 所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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9. If x is a positive integer, is x < 16? (1) x is less than the average (arithmetic mean) of the first ten positive integers. (2) x is the square of 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: x<16? (1) 即x< (1+2+…+10)/10=5.5時 所以可以判定，sufficient。 (2) 即x是整數平方時 x可以為32=9, 42=16, 52=25 所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------10. 錯 Of the 800 students at a certain college, 250 students live on campus and are more than 20 years old. How many of the 800 students live on campus and are 20 years old or less? (1) 640 students at the college are more than 20 years old. (2) 60 students at the college are 20 years old or less and live off campus. (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: 有多少人住校等於或小於20歲?

>20 <=20

campus 250 b

~ campus c d Total:800

5

11. 錯 If p is a positive odd integer, what is the remainder when p is divided by 4? (1) When p is divided by 8, the remainder is 5. (2) p is the sum of the squares of two positive integers. (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: p除以4的餘數? (1) 即p=8n+5=4(2n+1)+1時 餘數為1，所以可以判定，sufficient。 (2) 即p = x2 + y2時 又p為正奇數，所以x, y必為一奇一偶。 (i)所有偶數的平方都可以被4整除。 (ii)所有奇數的平方除以4，餘數都是1。 (2k+1)2=4k2+4k+1=4(k2+k)+1 餘數必為 0 + 1 = 1 所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------12. Of the 25 cars sold at a certain dealership yesterday, some had automatic transmission and some had antilock brakes. How many of the cars had automatic transmission but not antilock brakes? (1) All of the cars that had antilock brakes also had automatic transmission. (2) 2 of the cars had neither automatic transmission nor antilock brakes. (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: 有多少車有automatic transmission沒有antilock brakes? transmission

~ transmission

Brake

a

c

~Brake

b

d Total:25

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13. Is x to the right of -5 on the number line? (1) x is to the right of -7 on the number line. (2) x is between -4 and -3 on the number line. (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在數線上位於-5的右邊嗎? 即x > -5? (1) 即x位在-7的右邊時 => x>-7，可能在-5的左邊或右邊，所以無法判定，insufficient。 (2) 即x在-4和-3之間時 => -4 < x < -3，確定在-5的右邊，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------14. Is y between -2 and 1 on the number line? (1) y is to the right of -1 on the number line. (2) y is to the left of 2 on the number line. (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: 在數線上，y在-2和1的中間嗎? 即-2 < y < 1? (1) 即y位在-1的右邊時 => y>-1，有可能在-2和1之間，也有可能在1的右邊，所以無法判定，insufficient。 (2) 即y位在2的左邊時 => y<2，有可能在-2和1之間，也有可能在-2的左邊，所以無法判定，insufficient。 (3) 由(1)(2)合併，-1 < y < 2，還是無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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15. Is r to the right of -6 on the number line? (1) r is between -4 and -1 on the number line. (2) r is between -3 and 1 on the number line. (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: 在數線上，r在-6的右邊嗎? 即r > -6? (1) 即r在-4和-1之間時 => -4 < r < -1，確定在-6的右邊，所以可以判定，sufficient。 (2) 即r在-3和1之間時 => -3 < r < 1，確定在-6的右邊，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------16. Lines n and p lie in the xy-plane. Is the slope of line n less than the slope of line p? (1) Lines n and p intersect at the point (5,1). (2) The y-intercept of line n is greater than the y-intercept of line 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: 斜率n小於斜率p嗎? 直接畫圖最快 (1) 即相交於(5,1)時 資訊不足，所以無法判定，insufficient。 (2) 即直線n的y截距大於直線p的y截距時 資訊不足，無法判定，insufficient。 (3) 由(1)(2)合併，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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17. 錯 Six countries in a certain region sent a total of 75 representatives to an international congress, and no two countries sent the same number of representatives. Of the six countries, if Country A sent the second greatest number of representatives, did Country A send at least 10 representatives? (1) One of the six countries sent 41 representatives to the congress. (2) Country A sent fewer than 12 representatives to the congress. (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: A擁有第二多的代表數，則A至少有10位代表? (聰明點，其實只要討論A=9和10的情形是否可以成立) (1) 即f=41時，又a+b+c+d+e+f=75 a+b+c+d+e = 34 可知41為裡面最大值，又a, b, c, d, e, 41為相異 且a為第二大值，所以a在剩下的5個區域代表數有最大值 (i) a=9 時 => b+c+d+e=23 => (b, c, d, e) =(2, 6, 7, 8) (ii) a=10 時 => b+c+d+e=22 => (b, c, d, e) =(4, 5, 6, 9) => 都能成立，所以無法判定，insufficient。 (2) 即 a<12 時，又a+b+c+d+e+f=75，且a, b, c, d, e, f為相異 且a為第二大值 (i) a=9 時 => b+c+d+e+f=66 => (b, c, d, e, f) =(5, 6, 7, 8, 40) (ii) a=10 時 => b+c+d+e+f =65 => (b, c, d, e, f) =(4, 5, 6, 8, 42) => 都能成立，所以無法判定，insufficient。 (3) 由(1)(2)合併，即f=41時，a<12 時，又a+b+c+d+e=34，且a, b, c, d, e, f為相異 且a為第二大值 (i) a=9 時 => b+c+d+e=23 => (b, c, d, e) =(2, 6, 7, 8) (ii) a=10 時 => b+c+d+e=22 => (b, c, d, e) =(4, 5, 6, 9) => 都能成立，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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18. What is the hundredths digit of the decimal z? (1) The tenths digit of 100z is 2. (2) The units digit of 1,000z is 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】E 【思路】題目問: z的百分之一位數? (1) 即100z的十分位數為2時 可知z的千分之一位數為2，所以無法判定，insufficient。 (2) 即1,000z的個位數為2時 可知z的千分之一位數為2，所以無法判定，insufficient。 (3) 由(1)(2)合併，還是無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------19. Is z equal to the median of the three positive integers x, y, and z? (1) x < y + z (2) 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】B 【思路】題目問: z為x, y, z的中位數? (1) 即 x < y+z 時 直接代數字下去推 x = 1, y = 2, z = 3: z is not the median x = 1, y = 3, z = 2: z is the median 所以無法判定，insufficient。 (2) 即 y=z 時 表示三個數中有兩個數一樣，所以可以判定中位數，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

10

20. 錯 A certain one-day seminar consisted of a morning session and an afternoon session. If each of the 128 people attending the seminar attended at least one of the two sessions, how many of the people attended the morning session only? (1) 3/4 of the people attended both sessions. (2) 7/8 of the people attended the afternoon session. (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: 多少人只參加Morning session? morning

~ morning

afternoon

a

c

~ afternoon

b

d Total:128

11

22. Of the 800 sweaters at a certain store, 150 are red. How many of the red sweaters at the store are made of pure wool? (1) 320 of the sweaters at the store are neither red nor made of pure wool. (2)

100 of the red sweaters at the store are not made of pure wool.

(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: 多少紅色毛衣是由羊毛製成? 已知A=150，問A∩B=? (1) 即A’∩B’=320時 A’∩B’ = (A∪B)’ = 全 - (A∪B) = 全 - [A+B- (A∩B)] => 320 = 800 - [ 150 + B - (A∩B)] 不知道B，所以無法判定，insufficient。 (2) 即A∩B’=100時 A∩B’ = A - (A∩B) => 100 = 150 - (A∩B) => A∩B=50，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

12

23. 錯 At least 100 students at a certain high school study Japanese. If 4 percent of the students at the school who study French also study Japanese, do more students at the school study French than Japanese? (1) 16 students at the school study both French and Japanese. (2)

10 percent of the students at the school who study Japanese also study French.

(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: 修法語的學生多於修日語的學生? French

~ French

Japanese

a

c

~ Japanese

b

d Total:

13

25. What is the value of n? (1) n is between 0 and 1. (2) 7/16 is 3/8 more than 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值? (1) 即0<n<1時 不知道n，所以無法判定，insufficient。 (2) 即7/16 - n=3/8時 n=1/16，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------26. In a certain conference room each row of chairs has the same number of chairs, and the number of rows is 1 less than the number of chairs in a row. How many chairs are in a row? (1) There is a total of 72 chairs. (2)

After 1 chair is removed from the last row, there is a total of 17 chairs in the last 2 rows.

(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: 一排有多少張椅子? 設有r排，所求即問 r+1=? (1) 即r*(r+1)=72時 => (r+9)(r-8)=0，r=8，所以可以判定，sufficient。 (2) 即2(r+1) – 1 = 17時 => r=8，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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27. What is the price for a certain meal listed on a menu? (1) The total paid for the meal, sales tax, and gratuity is \$10.84. (2)

The sales tax on food is 6 percent.

(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: 餐點多少錢? (1) 即M+S+G=10.84時 不知道S和G，所以無法判定，insufficient。 (2) 即S=0.06M時 不知道M，所以無法判定，insufficient。 (3) 由(1)(2)合併，1.06M + G = 10.84 不知道G，所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------28. Which of Company X and Company Y earned the greater gross profit last year? (1) Last year the expenses of Company X were 5/6 of the expenses of Company Y. (2) Last year the revenues of Company X were \$6 million less than the revenues of Company 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和Y誰有較大淨收入? 即 Xr-Xe 和 Yr-Ye 誰有較大值? (1) 即Xe=(5/6)Ye時 只知道支出，不知道收入，所以無法判定，insufficient。 (2) 即Xr=Yr – 6,000,000時 只知道收入，不知道支出，所以無法判定，insufficient。 (3) 由(1)(2)合併 Xr-Xe= (Yr – 6,000,000) - (5/6)Ye 和Yr – Ye相比較 不知道Ye，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

15

29. What is the value of 6x - 10? (1) 3x - 5 = 16 (2)

12x - 10 = 74

(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: 6x - 10=? (1) 即3x - 5 = 16時 => x=7，所以可以判定，sufficient。 (2) 即12x – 10 = 74時 => x=7，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------30. Is w greater than 1? (1) w + 2 > 0 (2) w2 > 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: w>1? (1)即w>-2時，所以無法判定，insufficient。 (2)即w>1或w<-1時，所以無法判定，insufficient。 (3)由(1)(2)合併，即w>1或-2<w<-1時，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

16

31. At a refreshment stand, each can of soda sells for the same price and each sandwich sells for the same price. What is the total price for 2 sandwiches and 3 cans of soda at the stand? (1) At the stand the total price for 1 sandwich and 1 can of soda is \$3. (2)

At the stand the total price for 3 sandwiches and 2 cans of soda is \$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: 2個三明治，3瓶汽水價錢? 2x+3y=? (1)即x+y=3時，所以無法判定，insufficient。 (2)即3x+2y=8時，所以無法判定，insufficient。 (3)由(1)(2)合併，得到x=2，y=1，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------32. Al, Pablo, and Marsha shared the driving on a 1,500-mile trip. Which of the three drove the greatest distance on the trip? (1) Al drove 1 hour longer than Pablo but at an average rate of 5 miles per hour slower than Pablo. (2)

Marsha drove 9 hours and averaged 50 miles per hour.

(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: 誰開車行駛最大距離? A+P+M=1500 (1) 即A=VA*(t+1)和P=(VA+5)*t時 不知道Martha資訊，所以無法判定，insufficient。 (2) 即M=9*50=450時 不知道Al和Pablo資訊，所以無法判定，insufficient。 (3)由(1)(2)合併，不知道Al和Pablo行駛距離，所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------33. If x=1/2, is y equal to 1? (1) y2 ( x + 1/2) = 1 (2) y ( 2x – 1 ) = 2x – 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: y=1? (1)即y=1或-1時，所以無法判定，insufficient。 (2)即y=1時，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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34. 錯 During an experiment, some water was removed from each of 6 water tanks. If the standard deviation of the volumes of water in the tanks at the beginning of the experiment was 10 gallons, what was the standard deviation of the volumes of water in the tanks at the end of the experiment? (1) For each tank, 30 percent of the volume of water that was in the tank at the beginning of the experiment was removed during the experiment. (2) The average (arithmetic mean) volume of water in the tanks at the end of the experiment was 63 gallons. (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】題目問標準差?

(1)每個水桶被移走30%的水時，標準差會變成0.7倍，所以可以判定，sufficient。 (2)只知道平均數，但不知道變異程度，所以無法判定，insufficient。 (補充)標準差性質: (1)若項數同時增加或減少某數，則標準差不變。 (2)若項數同時增加或減少某比例，則標準差會等比增加或減少。 -------------------------------------------------------------------------------------------------------------------------------------------------------35. If p and n are positive integers and p > n, what is the remainder when p2 - n2 is divided by 15? (1) The remainder when p + n is divided by 5 is 1. (2)

The remainder when p - n is divided by 3 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】E 【思路】題目問: p2 - n2除以15的餘數? p2 - n2=(p+n) (p-n) (1) 即p+n=5a+1時 p2 - n2=(5a+1) (p-n) 資訊不足，所以無法判定，insufficient。 (2) 即p-n=3b+1時 p2 - n2=(p+n) (3b+1) 資訊不足，所以無法判定，insufficient。 (3)由(1)(2)合併，(p+n) (p-n) =(5a+1)(3b+1)=15ab+5a+3b+1，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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36. If x is positive, is x > 3? (1) (x - 1)2 > 4 (2)

(x - 2)2 > 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: x>3? (1)即x-1>2或x-1<-2時，x>3或x<-1(不合)，所以可以判定，sufficient。 (2)即x-2>3或x-2<-3時，x>5或x<-1(不合)，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------37. When 1,000 children were inoculated with a certain vaccine, some developed inflammation at the site of the inoculation and some developed fever. How many of the children developed inflammation but not fever? (1) 880 children developed neither inflammation nor fever. (2)

20 children developed fever.

(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: 多少孩童出現紅腫卻沒有發燒? inflammation

~ inflammation

fever

a

c

~ fever

b

d Total:

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38. 錯 If r and s are positive integers, is r/s an integer? (1) Every factor of s is also a factor of r. (2) Every prime factor of s is also a prime factor of 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: r/s是整數? (1) 每個s的因數也是r的因數時 所以s一定是r的因數，可以判定，sufficient。 (2) 每個s的質因數也是r的質因數時 但質因數個數無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------39. For the students in class A, the range of their heights is r centimeters and the greatest height is g centimeters. For the students in class B, the range of their heights is s centimeters and the greatest height is h centimeters. Is the least height of the students in class A greater than the least height of the students in class B? (1) r < s (2) g > h (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: g-r > h-s? (1) 即r<s時，不知道g, h關係，所以無法判定，insufficient。 (2) 即g>h時，不知道r, s關係，所以無法判定，insufficient。 (3)由(1)(2)合併，可知 g-r > h-s 恆成立，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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40. Beth's bank charges a service fee on a regular checking account for each month in which the balance on the account falls below \$100 at any time during the month. Did the bank charge a service fee on Beth's regular checking account last month? (1) During last month, a total of \$1,000 was withdrawn from Beth's regular checking account. (2)

At the beginning of last month, Beth's regular checking account balance was \$500.

(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: 上個月有被收取服務費? (1)不知道帳戶金額情形，所以無法判定，insufficient。 (2)不知道月底帳戶金額，所以無法判定，insufficient。 (3)由(1)(2)合併，還是無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------41. Machines X and Y work at their respective constant rates. How many more hours does it take machine Y, working alone, to fill a production order of a certain size than it takes machine X, working alone? (1) Machines X and Y, working together, fill a production order of this size in two-thirds the time that machine X, working alone, does. (2) Machine Y, working alone, fills a production order of this size in twice the time that machine X, working alone, does. (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: 設總工程為1，X的工作時間為Tx，Y的工作時間為Ty，問 Ty - Tx=? (1) 即1/(1/Tx +1/Ty) = 2/3(Tx)，所以2Tx=Ty，無法判定，insufficient。 (2) 即Ty = 2Tx，所以無法判定，insufficient。 (3)由(1)(2)合併，還是無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

21

42. 錯 John and Mary own shares of stock in a certain company. Does John own more shares of the company's stock than Mary? (1) Mary owns more than 500 shares of the company's stock. (2) The number of shares of the company's stock that John owns is 400 less than twice the number of shares of the company's stock that Mary owns. (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: John的股票比Mary多? (1)即M>500時，不知道John的情形，所以無法判定，insufficient。 (2)即J=2M - 400時，所以無法判定，insufficient。 (3)由(1)(2)合併，假設 M>J時 即M > 2M – 400 = > M < 400 又M > 500 => 假設錯誤 J會恆大於M，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------43. Each person in a certain group supports only one of the two candidates R and T. Of the people in the group, 45 percent support Candidate R and the rest support Candidate T. How many people in the group are in favor of a flat tax? (1) Of the people in the group who support Candidate R, 58 percent are in favor of a flat tax. (2)

Of the people in the group who support Candidate T, 22 are in favor of a flat tax.

(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: 多少人支持統一稅? (1)即0.45*0.58*P時，不知道Candidate T那邊的情形，所以無法判定，insufficient。 (2)即0.55*0.22*P時，不知道Candidate R那邊的情形，所以無法判定，insufficient。 (3)由(1)(2)合併，不知道總人數P的情形，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

22

44. In the xy-plane, at what two points does the graph of y = (x + a)(x + b) intersect the x-axis? (1) a + b = -1 (2) The graph intersects the y-axis at (0, -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: 直線是否相交在x軸? (1)即a+b=-1時，a不一定等於b，所以無法判定，insufficient。 (2)即(0, -6)代入=> a*b=-6時，a不一定等於b，所以無法判定，insufficient。 (3)由(1)(2)合併，(a, b)=(-3, 2)或(2, -3)，可以判定不相交，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------45. How many of the 42 people in a group are employed students? (1) 29 of the 42 people are employed. (2)

24 of the 42 people are students.

(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: 多少人半工半讀? A∩B=? (1)即A=29時，所以無法判定，insufficient。 (2)即B=24時，所以無法判定，insufficient。 (3)由(1)(2)合併，還是無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------46. 錯 How many hours did it take Helen to drive from her house to her parents' house? (1) Helen's average speed on this trip was 72 kilometers per hour. (2) If Helen's average speed on this trip had been 8 kilometers per hour greater, it would have taken her 1 hour less. (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: 開車花多少時間? (1)即知道平均速度，所求=d/72，但不知道d，所以無法判定，insufficient。 (2)即d/v - d/(v+8)=1時，所以無法判定，insufficient。 (3)由(1)(2)合併，又v=72，可解出d=720，t=720/72=10，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

23

47. Is m ≠ n? (1) m + n < 0 (2) mn < 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】B 【思路】題目問: m≨n? (1)即m+n<0時，可以為兩負，所以無法判定，insufficient。 (2)即mn<0時，必為一正一負，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------48. Last Friday each of the pets at a certain veterinary clinic was given either 1 treat or 2 treats. What was the total number of treats given to pets at the clinic last Friday? (1) The total number of pets at the clinic last Friday was 90. (2) 2/3 of the pets at the clinic last Friday were given 2 treats 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】C 【思路】題目問: 寵物吃了多少餐? 即x+2y=? (1)即x+y=90時，所以無法判定，insufficient。 (2)即2/3(x+y)=y時，=> 2x =y，所以無法判定，insufficient。 (3)由(1)(2)合併，x=30, y=60，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------49. Does x + y = 5? (1) 4x + y = 17 (2) x + 4y = 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: x+y=5? (1)即4x+y=17時，所以無法判定，insufficient。 (2)即x+4y=8時，所以無法判定，insufficient。 (3)由(1)(2)合併，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

24

50. 錯 A construction company was paid a total of \$500,000 for a construction project. The company's only costs for the project were for labor and materials. Was the company's profit for the project greater than \$150,000? (1) The company's total cost was three times its cost for materials. (2)

The company's profit was greater than its cost for labor.

(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: 利潤大於150,000? 是否Profit = 500,000- (L+M) > 150,000? (1)即L+M=3M時，L=2M，不知道M，所以無法判定，insufficient。 (2)即500,000- (L+M) > L時，所以無法判定，insufficient。 (3)由(1)(2)合併，500,000 – (2M + M) >2M，=> M < 100,000 所以3M < 300,000，可以判定Profit =500,000 – 3M會大於200,000，所以可以判定所求，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------51. What is the total value of Company H's stock? (1) Investor P owns 1/4 of the shares of Company H’s total stock. (2) The total value of Investor Q’s shares of Company H’s stock is \$16,000. (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: H公司股票總價值? (1) 即P有1/4的H公司股票時 所以無法判定，insufficient。 (2) 即Q有H公司股票價值16,000時 所以無法判定，insufficient。 (3)由(1)(2)合併，還是無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

25

52. 錯 If the average (arithmetic mean) of four different numbers is 30, how many of the numbers are greater than 30? (1) None of the four numbers is greater than 60. (2)

Two of the four numbers are 9 and 10, respectively.

(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: 多少個數字會大於30? 已知 w+x+y+z = 120 (1) 即w, x, y, z<60時 (w, x, y, z)=(31, 32, 29, 28) or (31, 32, 33, 24) 所以無法判定，insufficient。 (2) 即9, 10, y, z時，y+z=101， (y, z)=(31, 80) or (29, 72) 所以無法判定，insufficient。 (3)由(1)(2)合併，=> 9, 10, y, z，y+z=101，y, z<60 直接討論臨界點 (i) 令y=59 => z=42 (ii) 令y=31 => z=70(不合) (iii) 令y=29 => z=72(不合) (y, z)= (59,42)…. (42,59) 所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------53. If wx = y, what is the value of xy? (1) wx2 = 16 (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: xy=? (1)代入即xy=16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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54. Is x - y + 1 greater than 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】B 【思路】題目問: x-y+1 > x+y-1? 即y<1? (1) 即x>0時，不知道y，所以無法判定，insufficient。 (2) 即y<0時，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------55. Malik's recipe for 4 servings of a certain dish requires 1 1/2 cups of pasta. According to this recipe, what is the number of cups of pasta that Malik will use the next time he prepares this dish? (1) The next time he prepares this dish, Malik will make half as many servings as he did the last time he prepared the dish. (2) Malik used 6 cups of pasta the last time he prepared this dish. (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: 下次用多少杯麵糰? (1)即準備上次一半份量時，但上次準備多少不知道，所以無法判定，insufficient。 (2)即上次用6杯麵糰時，但這次用多少不知道，所以無法判定，insufficient。 (3)由(1)(2)合併，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------56. During a sale, a clothing store sold each shirt at a price of \$15 and each sweater at a price of \$25. Did the store sell more sweaters than shirts during the sale? (1) The average (arithmetic mean) of the prices of all of the shirts and sweaters that the store sold during the sale was \$21. (2) The total of the prices of all of the shirts and sweaters that the store sold during the sale was \$420. (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: 毛衣賣的比T-Shirt多? 問x > y? (1) 即(15x+25y)/(x+y)=21時 6x=4y，x:y = 2:3，所以可以判定，sufficient。 (2) 即15x+25y=420時 所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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57. 錯 During a sale, for each shirt that Mark purchased at the regular price, he also purchased a shirt at half the regular price. How many shirts did Mark purchase during the sale? (1) The regular price of each of the shirts that Mark purchased during the sale was \$21.50. (2) The total of the prices for all the shirts that Mark purchased during the sale was \$129.00. (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: 買了多少件Shirt? 問x=? 看清楚題目說明，每purchase一件regular price shirt的同時，也pruchase一件半價的 (1)即平時售價 p =21.5時，半價是10.75，不知道件數，所以無法判定，insufficient。 (2)即x * [p + (1/2)p] = 129時，不知道p，所以無法判定，insufficient。 (3)由(1)(2)合併，x *(3/2)* 21.5 = 129，=> x=4，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------58. What is the remainder when the positive integer x is divided by 3? (1) When x is divided by 6, the remainder is 2. (2) When x is divided by 15, the remainder is 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: x除以3的餘數? (1)即x=6a+2=3(2a)+2時，所以可以判定，sufficient。 (2)即x=15b+2=3(5b)+2時，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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59. Ann deposited money into two new accounts, A and B. Account A earns 5 percent simple annual interest and account B earns 8 percent simple annual interest. If there were no other transactions in the two accounts, then the amount of interest that account B earned in the first year was how many dollars greater than the amount of interest that account A earned in the first year? (1) Ann deposited \$200 more in account B than in account A. (2)

The total amount of interest that the two accounts earned in the first year was \$120.

(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: Account B的利息 - Account A的利息? 問 B*8% - A*5% = ? (1) 即 B - A=200 時 => B = A+200，不知道A，所以無法判定，insufficient。 (2) 即 B*8% + A*5% = 120 時 所以無法判定，insufficient。 (3)由(1)(2)合併，可解出 A,B，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------60. What was the percent increase in the population of City K from 1980 to 1990? (1) In 1970 the population of City K was 160,000 (2) In 1980 the population of City K was 20 percent greater than it was in 1970, and in 1990 the population of City K was 30 percent greater than it was in 1970. (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: 1980 to 1990年，人口增加比例? (1)即1970年有160,000人時，所以無法判定，insufficient。 (2)即1970年有x人時，1980有1.2x人，1990年有1.3x人時，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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61. 錯 Of the 60 animals on a certain farm, 2/3 are either pigs or cows.

How many of the animals are cows?

(1) The farm has more than twice as many cows as it has pigs. (2) The farm has more than 12 pigs. (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: 多少動物是牛? P+C = 60*(2/3) = 40，問C=? (1)即P>2C時，所以無法判定，insufficient。 (2)即P>12時，所以無法判定，insufficient。 (3)由(1)(2)合併，P+C=40 => 2C+C<40，C < 13.33， 又C>12，可知C=13，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------62. In isosceles  RST what is the measure of  R (1) The measure of

 T is 100°

(2)

 S is 40°

The measure of

(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: 角R=? (1) 即角T=100時 可以判定剩下兩角為等角，所以角R=40可以判定，sufficient。 (2) 即角S=40時 無法判定角R是否為等角，角R可能為40或(180-40)/2=70，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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63. The cost of delivery for an order of desk chairs was \$10.00 for the first chair, and \$1.00 for each additional chair in the order. If an office manager placed an order for n desk chairs, is n > 24? (1) The delivery cost for the order totaled more than \$30.00. (2) The average (arithmetic mean) delivery cost per chair of the n chairs was \$1.36. (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】B 【思路】題目問: 訂了超過24張椅子? 即問x+1>24? (1) 即10+(1*x)>30時 x>20 => x+1>21，所以無法判定，insufficient。 (2) 即(10+1*x)/(1+x)=1.36時 可以求出x值，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------64. 錯 Are at least 10 percent of the people in Country X who are 65 years old or older employed? (1) In Country X, 11.3 percent of the population is 65 years old or older. (2) In Country X, of the population 65 years old or older, 20 percent of the men and 10 percent of the women are employed. (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: 至少10%的65歲以上人口會被雇用? (1)不知employed資訊，所以無法判定，insufficient。 (2)假設65歲(含)以上男生數為M，女生數為W 思考：0.2M+0.1W恆大於0.1(M+W)，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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65. If x is a positive number less than 10, is z greater than the average (arithmetic mean) of x and 10? (1) On the number line, z is closer to 10 than it is to x. (2) z = 5x (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: z大於x和10的算術平均? 已知x < 10，求 z > (x+10)/2? (1) 即z比較靠近10，比較遠離x時，直接畫圖即可 0 x z 10 z -----|---------|------|---|---|------------所以可以判定，sufficient。 (2) 即z=5x時 10x > x+10 => x > 10/9，但x可能為0 < x < 10/9，所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------66. In the finite sequence of positive integers K1, K2, K3, . . . , K9, each term after the second is the sum of the two terms immediately preceding it. If K5 = 18, what is the value of K9? (1) K4 = 11 (2) K6 = 29 (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: 即Fibonacci數列，K9=? 已知K5=18 (1)即K4=11時，所以可以判定，sufficient。 (2)即K6=29時，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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67. Is x + y negative? (1) x 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+y<0 ? (1)即x<0時，不知道y，所以無法判定，insufficient。 (2)即y>0時，不知道x，所以無法判定，insufficient。 (3)由(1)(2)合併，不知道x+y正負，所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------68. What is the tens digit of the positive integer r? (1) The tens digit of r/10 is 3. (2) The hundreds digit of 10r 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: 正整數r的十位數=? (1) 即r/10的十位數為3時 可知r的百位數為3，所以無法判定，insufficient。 (2) 即10r的百位數為6時 可知r的十位數為6，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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69.

In triangle ABC above, what is the length of side BC? (1) Line segment AD has length 6. (2) x = 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】A 【思路】題目問: BC邊長=? 已知AD=BD=BC (1)即AD=6時，所以可以判定，sufficient。 (2)即x=36時，不知道任何邊長資訊，所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------70. Is the integer x divisible by 6? (1) x + 3 is divisible by 3. (2) x + 3 is an odd 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】C 【思路】題目問: 整數x可以被6整除? (1) 即x+3可以被3整除時 x+3=3a => x=3a-3，所以無法判定，insufficient。 (2) 即x+3是奇數時 x+3=2b+1 => x=2b-2，所以無法判定，insufficient。 (3)由(1)(2)合併，x=6c+6 = 6(c+1)，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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71.

In the figure above, if x and y are each less than 90 and PS∥QR is the length of segment PQ less than the length of segment SR ? (1) x > y (2) x + y > 90 (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: PQ小於SR? (1)即x>y時，SR>PQ，所以可以判定，sufficient。 (2)即x+y>90時，無法判定角度大小，所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------72. A bookstore that sells used books sells each of its paperback books for a certain price and each of its hardcover books for a certain price. If Joe, Maria, and Paul bought books in this store, how much did Maria pay for 1 paperback book and 1 hardcover book? (1)

Joe bought 2 paperback books and 3 hardcover books for \$12.50.

(2)

Paul bought 4 paperback books and 6 hardcover books for \$25.00.

(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】E 【思路】題目問: P+H=? (1)即2P+3H=12.5時，所以無法判定，insufficient。 (2)即4P+6H=25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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73. What is the average (arithmetic mean) of eleven consecutive 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個連續正整數的平均值? 也就是問a6=?，直接畫圖表示最快 (1)即a5=7時，可以推出a6=8，所以可以判定，sufficient。 (2)即a7=9時，可以推出a6=8，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------74. If x and y are integers, what is the value of x + y? (1)

690 < x < y < 696

(2)

692 < x < y < 695

(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+y=? (1) 即690 < x < y < 696時 不知道x, y，所以無法判定，insufficient。 (2) 即692 < x < y < 695時 x=693, y=694 所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------75. Is x less than 20? (1) The sum of x and y is less than 20. (2) y is less than 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】E 【思路】題目問: x<20? (1)即x+y<20時，不知道y的正負，所以無法判定，insufficient。 (2)即y<20時，y有可能為負，所以無法判定，insufficient。 (3)由(1)(2)合併，y有可能為負，所以還是無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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76. Is the integer k divisible by 4? (1)

8k is divisible by 16.

(2)

9k is divisible by 12.

(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能被4整除? (1) 即8k/16=23*k/24能整除時 所以k至少包含一個2，所以無法判定，insufficient。 (2) 即9k/12=32*k /22*31能整除時 所以k至少包含兩個2，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------77. If n is an integer between 10 and 99, is n < 80? (1) The sum of the two digits of n is a prime number. (2) Each of the two digits of n 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】B 【思路】題目問: n<80? 令n=10x+y，又10<n<99，=> 1<x<9，0<y<9 (1) 即x+y可能為2, 3, 5, 7, 11, 13, 17時 n可能為98或12，所以無法判定，insufficient。 (2) 即x, y={2, 3, 5, 7}時 最大值為77，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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78. For each customer, a bakery charges p dollars for the first loaf of bread bought by the customer and charges q dollars for each additional loaf bought by the customer. What is the value of p? (1) A customer who buys 2 loaves is charged 10 percent less per loaf than a customer who buys a single loaf. (2) A customer who buys 6 loaves of bread is charged 10 dollars. (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=? (1) 即p+q=2*p(1 - 10%)時 p+q=1.8p，所以無法判定，insufficient。 (2) 即p+5q=10時 所以無法判定，insufficient。 (3)由(1)(2)合併，p=2，q=1.6，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------79. If the terms of a sequence are t1, t2, t3, . . . , tn, what is the value of n? (1) The sum of the n terms is 3,124. (2) The average (arithmetic mean) of the n terms is 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: n=? (1) 即n項和=3,124時 所以無法判定，insufficient。 (2) 即n項的算術平均數=4時 所以無法判定，insufficient。 (3)由(1)(2)合併，4n=3,124 => n=781，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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80. If x is a negative number, what is the value of x? (1) x2 = 1 (2) x2 + 3x + 2 = 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: x=? 已知x <0 (1)即x=1(不合)或-1時，所以可以判定，sufficient。 (2)即x=-2或-1時，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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81.

A C

B

In the figure above, what is the ratio KN/MN (1)

The perimeter of rectangle KLMN is 30 meters.

(2)

The three small rectangles have the same dimensions.

(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: KN/MN? 設KN=x, MN=y，問x/y=? (1) 即2(x+y) =30時 不知道x, y，所以無法判定，insufficient。 (2) 即三個長方形的大小相等時

the three smaller triangles have same dimensions 注意，same dimensions是相同大小，而不是相同面積 => Mc=ab=KN, db=bc=cN 由共用邊可以得知db=bc=cN=(1/2)x 又矩形KNcd=(1/3)矩形KNML => x *cN =(1/3) (x * y) => cN = (1/3)y 所以cN=(1/2)x= (1/3)y => x/y = 2/3 所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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82. Is |x|>|y|? (1) x2 > y2 (2) 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】A 【思路】題目問: |x|>|y|? (1)即x2 > y2時 => |x|>|y|，所以可以判定，sufficient。 (2)即x > y時，不知道x, y的正負，所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------83. If u, v, and w are integers, is u > 0? (1) u = v2 + 1 (2) u = w4 + 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: u>0? (1) 即u=v2+1時，u一定>0，所以可以判定，sufficient。 (2) 即u=w4+1時，u一定>0，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------84. 錯 If p is a prime number greater than 2, what is the value of p? (1)

There are a total of 100 prime numbers between 1 and p + 1.

(2)

There are a total of p prime numbers between 1 and 3,912.

(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: p是大於2的質數，p=? (1) 即1和p+1之間有100個質數時 100個質數從2開始數起，第100個質數就是p 可以知道p的值，並不需要真的列出來 所以可以判定，sufficient。 (2) 即1到3,912有p個質數時 可以知道p的值，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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85. Is k positive? (1) k is between -2 and 3 on the number line. (2) k is between 1 and 2 on the number line. (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>0? (1)即-2 < k < 3時，所以無法判定，insufficient。 (2)即1<k<2時，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------86. 錯 Each of the students in a certain class received a single grade of P, F, or I. What percent of the students in the class were females? (1) Of those who received a P, 40 percent were females. (2) Of those who received either an F or I, 80 percent were males. (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: 女生佔全班人數多少比率? (1)即P中有40%女生時，不知道F和I的女生比率，所以無法判定，insufficient。 (2)即F和I中有20%女生時，不知道P的女生比率，所以無法判定，insufficient。 (3)由(1)(2)合併，只知道比率，還是不知道P, I, F人數分布情形，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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87.

In the figure above, segments PQ and PR are each parallel to one of the rectangular coordinate axes. What is the sum of the coordinates of point P? (1)

The x-coordinate of point Q is -1.

(2)

The y-coordinate of point R 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】C 【思路】題目問: P點的座標總和? 即P(x,y)，求x+y=? (1)即x=-1時，不知道y，所以無法判定，insufficient。 (2)即y=1時，不知道x，所以無法判定，insufficient。 (3)由(1)(2)合併，x+y=0，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------88. 錯 What is the greatest common divisor of positive integers m and n? (1) m is a prime number. (2) 2n = 7m (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: (m, n)=? (1) 即m是質數時 n有可能是大於m或小於m 最大公因數可能為m或n 所以無法判定，insufficient。 (2) 即2n=7m時 m:n=2:7，所以無法判定，insufficient。 (3) 由(1)(2)合併，因為m, n 為正整數，又m為質數，所以m必為2才會有整數解 => n=7，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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89. What is the retail price of a certain calculator? (1) The retail price of the calculator is \$2.00 more than the wholesale price. (2) The retail price of the calculator is 50 percent more than the \$4.00 wholesale 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】B 【思路】題目問: 計算機的零售價? 問R=? (1)即R - S=2時，所以無法判定，insufficient。 (2)即R=6時，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------90. 錯 On Jane's credit card account, the average daily balance for a 30-day billing cycle is the average (arithmetic mean) of the daily balances at the end of each of the 30 days. At the beginning of a certain 30-day billing cycle, Jane's credit card account had a balance of \$600. Jane made a payment of \$300 on the account during the billing cycle. If no other amounts were added to or subtracted from the account during the billing cycle, what was the average daily balance on Jane's account for the billing cycle? (1) Jane's payment was credited on the 21st day of the billing cycle. (2) The average daily balance through the 25th day of the billing cycle was \$540. (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: 算帳戶在30天的付款週期下，每天餘額的算術平均數。 最初時帳戶有600塊，然後這三十天唯一變化就是某天支出了300。 (1) 即300元是在第21天中付出時 所以可知1~20天每日餘額是600，21~30天每日餘額是300， 平均值=(600*20+300*10)/(20+10)，所以可以判定，sufficient。 (2) 即前25天的餘額算術平均數=540時 表示支出300已經發生了，所以26~30天每日餘額是300 平均值=(540*25+300*5)/(25+5)，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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91. What is the remainder when the positive integer n is divided by 6? (1) n is a multiple of 5. (2) n is a multiple of 12. (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6的餘數? (1)即n=5a時，所以無法判定，insufficient。 (2)即n=12b=6(2b)時，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------92. Is the integer r divisible by 3? (1) r is the product of 4 consecutive positive integers. (2) r < 25 (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可以被3整除? (1) 即r=連續四個正整數乘積時 因為每三個連續正整數就有一個會是3的倍數，所以可以判定，sufficient。 (2) 即r<25時，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

45

93. 錯 If the integer n is greater than 1, is n equal to 2? (1) n has exactly two positive factors. (2) The difference of any two distinct positive factors of 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】B 【思路】題目問: 若n大於1，n等於2? (1) 即n只有兩個正因數時 即n就是質數，n=2, 3, 5, 7, 11...，所以無法判定，insufficient。 (2) 即任兩不同正因數的差為奇數時 又要滿足1是正因數的情形 表示另一正因數必為偶數。 (i) 討論另一正因數為2時 因數可能為1, 2 => n=2 (ii) 討論另一正因數為4時 因數可能為1, 2, 4 => 出現 4 - 2= 偶數 (不合) 所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------94. What is the value of y? (1) y is an odd integer between 28 and 34. (2) 31 < y < 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】C 【思路】題目問: y值? (1) 即y是28 < y < 34的奇數時 y可能為29, 31, 33，所以無法判定，insufficient。 (2) 即31 < y < 36時 所以無法判定，insufficient。 (3)由(1)(2)合併，y=33，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

46

95. Is 1/(a-b) < b - a? (1) a < b. (2) 1 < |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】A 【思路】題目問: 1/(a-b) < b-a? (1) 即a < b時 => 1/(a-b)<0，b-a > 0 =>原式恆成立 所以可以判定，sufficient。 (2) 即1<|a-b|時 => 即a-b > 1 或 a-b < -1 (i) a-b > 1時： 1/(a-b) > 0，b-a < -1 =>原式恆不成立 (ii) a-b <-1時： 1/(a-b) < 0，b-a > 1 =>原式恆成立 所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------96. If x and y are integers greater than 1, is x a multiple of y? (1) 3y2 + 7y = x. (2) x2 – x is a multiple of 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】A 【思路】題目問: x是y的倍數? 典型的AT文字敘述陷阱。 (1)即x=y(3y+7)時，所以可以判定，sufficient。 (2)即x2 – x=ky時，x(x-1)=ky，可能x=y或x-1=y，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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97. Is sqrt[(x-3)2]= 3–x ? (1) x ≠ 3 (2) –x|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】B 【思路】題目問: sqrt[(x-3)2]= 3–x? 原式成立時，x-3<0，即x<3。 (1) 即x不等於3時 所以無法判斷，insufficient。 (2) 即-x>0，=> x<0 所以可以判斷，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------98. What is the value of 5x2 + 4x - 1? (1) x(x + 2) = 0 (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】B 【思路】題目問: 5x2 + 4x – 1=? (1)即x=0或-2時，所求=-1或11，所以無法判定，insufficient。 (2)即x=0時，所求=-1，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------99. If m > 0 and n > 0, is (m+x)/(n+x) > m/n? (1) m < n (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: (m+x)/(n+x) > m/n? 即分母分子同加一個數，直接代數字下去判斷最快 (1)即m<n時，但不知道x的正負，所以無法判定，insufficient。 (2)即x>0時，不知道m, n大小狀況，所以無法判定，insufficient。 (3)由(1)(2)合併，(m+x)/(n+x) > m/n，恆成立，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

48

100.

In the rectangular coordinate system shown above, does the line k (not shown) intersect quadrant II? (1) The slope of k is -1/6 (2) The y-intercept of k 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】A 【思路】題目問: 直線k會和第二象限相交? 直接畫圖比較好作答。 (1)即直線k斜率為-1/6時，會通過第二象限，所以可以判定，sufficient。 (2)即直線y截距為-6時，所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------101. If the sum of three different numbers is 54, what is the largest number? (1) The largest number is twice the smallest number. (2) The sum of the two smaller numbers 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】B 【思路】題目問: 三個不同數字的最大值=? 已知x+y+z=54，假設x<y<z，求z=? (1)即z=2x時，所以無法判定，insufficient。 (2)即x+y=30時，最大值z=24，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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102. If M is a finite set of negative integers, is the total number of integers in M an odd number? (1) The product of all the integers in M is odd. (2) The product of all the integers in M 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】B 【思路】題目問: 集合M的數字有奇數個? 已知集合M為負整數的有限集合 (1)即所有整數的乘積為奇數時，只能判斷集合M裡面的數字全是奇數，但不知道個數，所以無法判定，insufficient。 (2)即所有整數的乘積為負時，知道必為奇數個，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------103. A certain bag contains red, blue, and green marbles. What is the ratio of the number of green marbles to the number of red marbles in the bag? (1) The number of blue marbles in the bag is 2 times the number of green marbles in the bag. (2) The number of blue marbles in the bag is 3 times the number of red marbles in the bag. (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: G : R=? (1)即B=2G時，所以無法判定，insufficient。 (2)即B=3R時，所以無法判定，insufficient。 (3)由(1)(2)合併，G : R = 3 : 2，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------104. What is the value of x? (1) y - x = y - 6 (2) x + 2y = 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】A 【思路】題目問: x=? (1)即x=6時，所以可以判定，sufficient。 (2)即x+2y=10時，不知道y，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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105. What is the number of members of Club X who are at least 35 years of age? (1) Exactly 3/4 of the members of Club X are under 35 years of age. (2) The 64 women in Club X constitute 40 percent of the club’s membership. (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: 俱樂部有多少成員至少35歲? (1) 即1/4成員至少35歲時 只知道比例，不知道人數，所以無法判定，insufficient。 (2) 即總人數60/0.4=160人時 只知道人數，不知道年齡分佈，所以無法判定，insufficient。 (3)由(1)(2)合併，160*(1/4)=40，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------106. What was Jean's insurance premium in 1995? (1) The ratio of Jean’s insurance premium in 1995 to her insurance premium in 1994 was 6/5. (2) Jean’s insurance premium in 1995 was 20 percent more than her insurance premium in 1994. (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: 1995年的保險費? (1)只知道比例，不知道費用，所以無法判定，insufficient。 (2)只知道比例，不知道費用，所以無法判定，insufficient。 (3)由(1)(2)合併，只知道比例，不知道費用，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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107. A combined total of 55 light bulbs are stored in two boxes; of these, a total of 7 are broken. If there are exactly 2 broken bulbs in the first box, what is the number of bulbs in the second box that are not broken? (1) In the first box, the number of bulbs that are not broken is 15 times the number of broken bulbs. (2) The total number of bulbs in the first box is 9 more than the total number of bulbs in the second box. (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: 第二個盒子有多少燈泡沒壞掉?

Box 1 Box 2

Good x

Broken 2

y

5

48

7

Total:55

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109. If R = 1 + 2xy + x2y2, what is the value of xy? (1) R = 0 (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】A 【思路】題目問: xy=? (1)即(xy+1)2=0時，xy=-1，所以可以判定，sufficient。 (2)即x>0時，不知道y，所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------110. If zn = 1, what is the value of z? (1) n is a nonzero integer. (2) 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】C 【思路】題目問: z=? (1)即n不等於0時，z=+1或-1，所以無法判定，insufficient。 (2)即z>0時，不知道n，所以無法判定，insufficient。 (3)由(1)(2)合併，z=1，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------111. If x, y, and z are positive integers, what is the remainder when 100x + 10y + z is divided by 7? (1) y = 6 (2) z = 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: 100x+10y+z除以7的餘數? (1)即y=6時，不知道x, z，所以無法判定，insufficient。 (2)即z=3時，不知道x, y，所以無法判定，insufficient。 (3)由(1)(2)合併，100x+10*6+3=7(14x+8)+2x+4+3=7(14x+8+1)+2x， 不知道x，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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114. 錯 Henry purchased 3 items during a sale. He received a 20 percent discount off the regular price of the most expensive item and a 10 percent discount off the regular price of each of the other 2 items. Was the total amount of the 3 discounts greater than 15 percent of the sum of the regular prices of the 3 items? (1) The regular price of the most expensive item was \$50, and the regular price of the next most expensive item was \$20. (2) The regular price of the least expensive item was \$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】A 【思路】題目問: 3個折扣的優惠大於15%平常價錢? 即0.2*X1 + 0.1*X2 + 0.1*X3 > 0.15*(X1+X2+X3) (1) 即X1=50，X2 =20時 折扣: 0.2*50 + 0.1*20 + 0.1*X3= 12 + 0.1 X3 15%: 0.15*(X1+X2+X3) = 0.15*(50+20+X3) =10.5 + 0.15 X3 若12 + 0.1 X3 > 10.5 + 0.15 X3 => X3 < 30 又X3是三者中價格最低者，所以X3 < 20，關係恆成立 所以可以判定，sufficient。 (2) 即X3=15時 不知道X1和X2，所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------115. If xy = -18, is x less than y? (1) x < 0 (2) y < 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】A 【思路】題目問: x<y? 已知 xy=-18 (1)即x<0時，=> y>0，所以可以判定，sufficient。 (2)即y<10時，但y可能為正或負，x可能為負或正，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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116. An insurance company has a contract with a medical laboratory to pay a discounted price for a certain medical test performed on patients referred to the laboratory by the insurance company. If the laboratory's original bill for this medical test on a patient referred by the insurance company is \$230, what is the percent discount specified by the contract between the laboratory and the insurance company? (1) The insurance company is required to pay only 20 percent of the original bill for the test. (2) The insurance company is required to pay \$46 for the test. (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: 折扣在交易中所佔的比例? (1)即只要付20%的original bill時 折扣就佔了80%，所以可以判定，sufficient。 (2)即後來只要付\$46時 折扣就佔了(230 - 46)/230=80%，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------117. If P, Q, and R are points on the number line, what is the distance between P and R? (1) Q is between P and R. (2) The distance between P and Q 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】E 【思路】題目問: P點和R點的距離? (1)即Q在P和R之間時，只知道相對位置，不知道距離，所以無法判定，insufficient。 (2)即|P - Q|=5時，只知道PQ距離，不知道R的資訊，所以無法判定，insufficient。 (3)由(1)(2)合併，還是無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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118. If x is positive, what is the value of y? (1) 5x = 15 (2) xy + y = 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: y=? (1)即x=3時，不知道y，所以無法判定，insufficient。 (2)即y(x+1)=18時，不知道x，所以無法判定，insufficient。 (3)由(1)(2)合併，x=3，y=9/2，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------119. A contractor combined x tons of a gravel mixture that contained 10 percent gravel G, by weight, with y tons of a mixture that contained 2 percent gravel G, by weight, to produce z tons of a mixture that was 5 percent gravel G, by weight. What is the value of x? (1) y = 10 (2) z = 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: x=? 混合物的問題，x噸10%砂石與y噸2%砂石混合，變成了z噸5%砂石。 其中x+y=z，x*10%+y*2%=z*5%=>10x+2y=5z，問x=? (1) 即y=10時 因為x+y=z，10x+2y=5z =>5x+5y=10x+2y => 5x=3y 所以可以判定，sufficient。 (2) 即z=16時 因為x+y=16，10x+2y=80 => x=6，y=10，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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120. How many of the students in a certain class are taking both a history and a science course? (1) Of all the students in the class, 50 are taking a history course. (2) Of all the students in the class, 70 are taking a science course. (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: 多少人同時修歷史和科學? (1)即50個人修歷史課時，所以無法判定，insufficient。 (2)即70個人修科學課時，所以無法判定，insufficient。 (3)由(1)(2)合併，所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------121. Pat bought 5 pounds of apples. How many pounds of pears could Pat have bought for the same amount of money? (1) One pound of pears costs \$0.50 more than one pound of apples. (2) One pound of pears costs 1 1/2 times as much as one pound of apples. (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: 一樣的錢可以買多少磅梨子? 即5*a = x*p，問x=? (1) 即p=a+0.5時 5 * a = x *(a+0.5) 不知道a，所以無法判定，insufficient。 (2) 即p=1.5a時， 5 * a = x * 1.5a，x=10/3，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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122. During a 40-mile trip, Marla traveled at an average speed of x miles per hour for the first y miles of the trip and at an average speed of 1.25x miles per hour for the last 40 - y miles of the trip. The time that Marla took to travel the 40 miles was what percent of the time it would have taken her if she had traveled at an average speed of x miles per hour for the entire trip? (1) x = 48 (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】B 【思路】題目問: 全程40時間和全程速度用每小時x英哩的時間比例? 第一段: y/x，第二段: (40-y)/1.25x，全程: 40/x 所求 = 左:右 = [y/x +(40-y)/1.25x] : [40/x] (1) 即x=48時 左 = y/48 + (40-y)/60 = (y+160)/240 右 = 40/48 不知道y，所以無法判定，insufficient。 (2) 即y=20時 左 = 20/x + (40-20)/1.25x = 36/x 右 = 40/x 所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------123. If integer p is greater than 1, is p a prime number? (1) p is odd. (2) The only positive factors of p are 1 and 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】B 【思路】題目問: p是質數? 已知p>1 (1)即p是奇數時，所以無法判定，insufficient。 (2)即質數定義，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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124. A certain company divides its total advertising budget into television, radio, newspaper, and magazine budgets in the ratio of 8:7:3:2, respectively. How many dollars are in the radio budget? (1) The television budget is \$18,750 more than the newspaper budget. (2) The magazine budget is \$7,500. (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: radio的預算? 已知 t : r : n : m = 8 : 7 : 3 : 2，問r=? (1)即 t - n = 18,750時，r = 18,750 * [7/(8-3)] = 26,250，所以可以判定，sufficient。 (2)即 m=7,500時，r=7,500 * (7/2) = 26,250，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------125. The sequence a1, a2, a3, . . . , an of n integers is such that ak = k if k is odd and ak = -ak - 1 if k is even. Is the sum of the terms in the sequence positive? (1) n is odd. (2) an 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】D 【思路】題目問: 序列總和為正? 即a1=1，a2=-a1=-1，a3=3，a4=-a3=-3 (1)即n是奇數時，所以可以判定為正，sufficient。 (2)即an為正時，所以可以判定為正，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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126. S is a finite set of numbers. Does S contain more negative numbers than positive numbers? (1) The product of all the numbers in S is -1,200. (2) There are 6 numbers 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的負數比正數多? (1) 即所有數字的乘積為-1,200時 只知道至少有一個負數，所以無法判定，insufficient。 (2) 即集合S有6個數字時 不知道集合內數字正負情形，所以無法判定，insufficient。 (3)由(1)(2)合併，還是無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------127. On the number line, if the number k is to the left of the number t, is the product kt to the right of t? (1) t < 0 (2) k < 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: kt在t的右邊? 即t<kt? 已知k在t的左邊，k<t，問t<kt? (1)即t<0時 => k<0 => kt>0，所以可以判定，sufficient。 (2)即k<1時，不知道k的正負，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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128. The points A, B, C, and D are on a number line, not necessarily in that order. distance between C and D is 8, what is the distance between B and D?

If the distance between A and B is 18 and the

(1) The distance between C and A is the same as the distance between C and B. (2) A is to the left of D on the number line. (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: BD距離=? 直接畫圖討論 (1)即CA=CB時，所以無法判定，insufficient。 (2)即A在D的左邊時，所以無法判定，insufficient。 (3)由(1)(2)合併，還是無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------129. Is x+1/x-3 < 0? (1) -1 < x < 1 (2) x2 - 4 < 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: x+1/x-3 < 0? 直接在數線上討論 (+) -1 (-) 3 (+) ------------|-----------------|-------(1) 即-1<x<1時，恆小於0，所以可以判定，sufficient。 (2) 即-2<x<2時，所以無法判定正負，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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130. In the sequence of positive numbers x1, x2, x3, . . . , what is the value of x1 ? (1) xn = (xn-1) /2，for all integers n > 1 (2) x5 = (x4) /(x4+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: x1=? (1) 即xn = (xn-1) /2時 為等比數列，但不知道值，所以無法判定，insufficient。 (2) 即x5 = (x4) /(x4+1) 時 不知道x1關係式，所以無法判定，insufficient。 (3)由(1)(2)合併，x5 = (x4) /2 = (x4) /(x4+1)，所以x4=1。 => x4= (x3) /2 => x3=2 => x3= (x2) /2 => x2=4 => x2= (x1) /2 => x1=8 所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------131. The positive integer k has exactly two positive prime factors, 3 and 7. If k has a total of 6 positive factors, including 1 and k, what is the value of k? (1) 32 is a factor of k. (2) 72 is not a factor of 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】D 【思路】題目問: k=? 已知k=3x * 7y，(x+1)(y+1)=6，x, y≧1 (1) 即32是k的因數時 => x至少有兩個3， => x≧2 =>(x,y)=(2,1)，所以可以判定，sufficient。 (2) 即72不是k的因數時 => k只有一個7 => y=1 => (x,y)=(2,1)，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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132. If l and k are lines in the xy-plane, is the product of the slopes of l and k equal to -1? (1) Line l passes through the origin and the point (1, 2). (2) Line k has x-intercept 4 and y-intercept 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】C 【思路】題目問: (直線l斜率) * (直線k斜率) =-1? (1) 即直線l通過(0,0), (1,2)時 直線l斜率=2，但不知道直線k狀況，所以無法判定，insufficient。 (2) 即直線k的x截距=4，y截距=2時， => 即通過(4,0), (0,2)兩個點 => 直線k斜率=-1/2，但不知道直線l狀況，所以無法判定，insufficient。 (3)由(1)(2)合併，直線l斜率=2，直線k斜率=-1/2， => 2*(-1/2)=-1，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------133. At a certain company, the average (arithmetic mean) number of years of experience is 9.8 years for the male employees and 9.1 years for the female employees. What is the ratio of the number of the company's male employees to the number of the company's female employees? (1) There are 52 male employees at the company. (2) The average number of years of experience for the company's male and female employees combined is 9.3 years. (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: 男女員工人數比=? M:F=? (1)即M=52時，不知道女生人數，所以無法判定，insufficient。 (2)即(9.8M+9.1F)/(M+F)=9.3時 =>0.5M = 0.2F，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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134. An attorney charged a fee for estate planning services for a certain estate. The attorney's fee was what percent of the assessed value of the estate? (1) The assessed value of the estate was \$1.2 million. (2) The attorney charged \$2,400 for the estate planning services. (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: 律師費和房地產價值的比例=? (1)即E=1,200,000時，不知道律師費，所以無法判定，insufficient。 (2)即A=2,400時，不知道房地產價值，所以無法判定，insufficient。 (3)由(1)(2)合併，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------135. 錯 During a one-day sale, a store sold each sweater of a certain type for \$30 more than the store's cost to purchase each sweater. How many of these sweaters were sold during the sale? (1) During the sale, the total revenue from the sale of these sweaters was \$270. (2) During the sale, the store sold each of these sweaters at a price that was 50 percent greater than the store's cost to purchase each sweater. (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: 在銷售中賣了多少件毛衣=? Revenue - cost = profit. (1)即(p+30)*n=270時，所以無法判定，insufficient。 (2)即p+30=1.5p時，p=60，還是不知道件數，所以無法判定，insufficient。 (3)由(1)(2)合併，n=3，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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136. 錯 An integer greater than 1 that is not prime is called composite. If the two-digit integer n is greater than 20, is n composite? (1) The tens digit of n is a factor of the units digit of n. (2)

The tens digit of n is 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: 大於20的二位數n是composite? 直接討論實際情形即可 (1) 22, 24, 26, 28, 33, 36, 39, 44, 48, 55, 66, 77, 88, 99，所以可以判定，sufficient。 (2) 21, 22, 23, 24, 25, 26, 27, 28, 29，其中23, 29為質數，所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------137. If x is an integer, is (x2 + 1)(x + 5) an even number? (1) x is an odd number. (2)

Each prime factor of x2 is greater than 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: (x2+1)(x+5)是偶數? (1)即x是奇數時 x+5必為偶數，所以可以判定，sufficient。 (2)x2的質因數皆大於7時 表示x2必為奇數, x 的質因數皆大於7 x2+1, x+5必為偶數，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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138. 錯 A certain theater has a total of 884 seats, of which 500 are orchestra seats and the rest are balcony seats. When tickets for all the seats in the theater are sold, the total revenue from ticket sales is \$34,600. What was the theater's total revenue from ticket sales for last night's performance? (1) The price of an orchestra seat ticket is twice the price of a balcony seat ticket. (2) For last night's performance, tickets for all the balcony seats were sold, but only 80 percent of the tickets for the orchestra seats were sold. (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: 昨晚演出的門票總收入=? 已知500*O+384*B=34,600 (1) 即O=2B時 完全沒有提到昨晚演出的門票銷售狀況，所以無法判定，insufficient。 (2) 即400*O+384*B時 不知道O, B票價資訊，所以無法判定，insufficient。 (3)由(1)(2)合併，B=25，O=50， 所求=400*50+384*25=29,600，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------139. If it took Carlos 1/2 hour to cycle from his house to the library yesterday, was the distance that he cycled greater than 6 miles? (Note: 1 mile = 5,280 feet) (1) The average speed at which Carlos cycled from his house to the library yesterday was greater than 16 feet per second. (2) The average speed at which Carlos cycled from his house to the library yesterday was less than 18 feet per second. (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: 來回一趟距離超過6英哩? (1)即速度超過(16/5,280)*60*30=5.45英哩時，所以無法判定，insufficient。 (2)即速度小於(18/5,280)*60*30=6.14英哩時，所以無法判定，insufficient。 (3)由(1)(2)合併，還是無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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140. At a certain pet shop, 1/3 of the pets are dogs and 1/5 of the pets are birds. How many of the pets are dogs? (1) There are 30 birds at the pet shop. (2) There are 20 more dogs than birds at the pet shop. (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: 多少寵物是狗? 已知D:B = 1/3: 1/5 = 5:3，問D=? (1)即B=30時，D=30*(1/3)*5=50，所以可以判定，sufficient。 (2)即D-B=20時，D=20*[5/(5-3)]=50，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------141. To fill an order on schedule, a manufacturer had to produce 1,000 tools per day for n days. What is the value of n? (1) Because of production problems, the manufacturer produced only 600 tools per day during the first 5 days. (2) Because of production problems, the manufacturer had to produce 1,500 tools per day on each of the last 4 days in order to meet the schedule. (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: n=? (1)即1,000*n = 600*5 + x*(n-5)時，不知道x，所以無法判定，insufficient。 (2)即1,000*n = 1,500*4 + y*(n-4)時，不知道y，所以無法判定，insufficient。 (3)由(1)(2)合併，還是無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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142. If x is a positive integer, what is the least common multiple of x, 6, and 9? (1) The least common multiple of x and 6 is 30. (2) The least common multiple of x and 9 is 45. (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, 6, 9]=? (1) 即[x, 6]=30時 [x, 6, 9]=[30, 9]=90 所以可以判定，sufficient。 (2) 即[x, 9]=45時 [x, 6, 9]=[45, 6]=90 所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------143.

In the figure shown, the measure of angle PRS is how many degrees greater than the measure of angle PQR? (1) The measure of angle QPR is 30° (2) The sum of the measures of angles PQR and PRQ is 150° (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PRS - 角PQR=? (1) 即角QPR=30度時 角PRS-角PQR=角QPR=30，所以可以判定，sufficient。 (2) 即角PQR+角PRQ=150度時 => 角QPR=30度，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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144. If \$1,000 is deposited in a certain bank account and remains in the account along with any accumulated interest, the dollar amount of interest, I , earned by the deposit in the first n years is given by the formula I =1000〔(1 + r/100)n - 1〕 , where r percent is the annual interest rate paid by the bank. Is the annual interest rate paid by the bank greater than 8 percent? (1) The deposit earns a total of \$210 in interest in the first two years. (2) ( 1 + r/100 )2 > 1.15 (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】A 【思路】題目問: 年利率大於8%? (1)即頭兩年賺了利息210元時 210 =1000[(1 + r/100)2 – 1] (1 + r/100)2 =1.21，r=10 所以可以判定，sufficient。 (2)因為(1 + 8/100)2 =1.1664 所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------145. 錯 On her way home from work, Janet drives through several tollbooths. Is there a pair of these tollbooths that are less than 10 miles apart? (1) The first tollbooth and the last tollbooth are 25 miles apart. (2) Janet drives through 4 tollbooths on her way home from work. (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: 有收費站間隔會小於10英哩? 看清楚題目的問法 (1) 即第一個與最後一個收費站的距離是25英哩時 但不知道中間到底有幾個收費站 所以無法判定，insufficient。 (2) 即知道有4個收費站時 4個收費站有3個間隔 但不知道距離資訊，所以無法判定，insufficient。 (3)由(1)(2)合併，4個收費站有3個間隔，25/3=8.33 表示至少有一個間距少於10英哩 所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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146. If the symbol △ represents either addition or multiplication, which operation does it represent? (1) a △ b = b △ a for all numbers a and b. (2) a △ ( b – c ) = ( a △ b ) – ( a △ c )for all numbers a, b, and 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】B 【思路】題目問: 三角形代表什麼運算符號? (1)△可表示*或+，所以無法判定，insufficient。 (2)△可表示*，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------147. 錯 Each employee on a certain task force is either a manager or a director. What percent of the employees on the task force are directors? (1) The average (arithmetic mean) salary of the managers on the task force is \$5,000 less than the average salary of all employees on the task force. (2) The average (arithmetic mean) salary of the directors on the task force is \$15,000 greater than the average salary of all employees on the task force. (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: 指導者在任務中所佔的人員比例? 令Manager人數=x，Director人數=y， Manger平均工資=m，Director平均工資=d 問y/(x+y) = ? (1) 即m - 5,000 = (mx+dy)/(x+y)時 所以無法判定，insufficient。 (2) 即d + 15,000 = (mx+dy)/(x+y)時 所以無法判定，insufficient。 (3)由(1)(2)合併，m - 5,000 = d+15,000 =>m = d+20,000 代入方程式 因為 d+15,000 = [(d+20,000)x+dy]/(x+y)，  dx + 15,000x + dy + 15,000y = dx + 20,000x + dy  15,000x + 15,000y = 20,000x  x/(x+y)=3/4， 所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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148. In the xy-plane, does the line with equation y = 3x + 2 contain the point (r, s)? (1) (3r + 2 - s)(4r + 9 - s) = 0 (2) (4r - 6 - s)(3r + 2 - s) = 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: y=3x+2通過(r, s)? 問3r-s + 2=0? (1)即3r+2-s=0或4r+9-s=0時，所以無法判定，insufficient。 (2)即4r-6-s=0或3r+2-s=0時，所以無法判定，insufficient。 (3)由(1)(2)合併，即3r+2-s=0，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------149. If m, r, x, and y are positive, is the ratio of m to r equal to the ratio of x to y ? (1) The ratio of m to y is equal to the ratio of x to r. (2) The ratio of m + x to r + y is equal to the ratio of x to 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: m : r=x : y? 即xr=my? (1) 即m : y=x : r時 => xy=mr，所以無法判定，insufficient。 (2) 即m+x : r+y=x : y時 => xr+xy=my+xy => xr=my，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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150. If a, b, k, and m are positive integers, is ak a factor of bm? (1) a is a factor of b. (2) k ≦ m (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: ak是bm的因數? (1)即a是b的因數時，所以無法判定，insufficient。 (2)即k≦m時，不知道a, b關係，所以無法判定，insufficient。 (3)由(1)(2)合併，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------151. Is x > 0.05? (1) x > 3/40 (2) x is greater than 3 percent of 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】D 【思路】題目問: x>0.05 ? 即x>1/20? (1)即x>3/40時，所以可以判定，sufficient。 (2)即x>50*3%=1.5時，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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152. If xyz > 0, is x > 0? (1) xy > 0 (2) xz > 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? 已知xyz>0 (1)即xy>0時 => z>0，所以無法判定，insufficient。 (2) xz>0時 => y>0，所以無法判定，insufficient。 (3)由(1)(2)合併，可知y>0, z>0 => x>0，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------153. A wooden rod is cut into two pieces. What is the length of the longer piece? (1) One of the pieces is 20 inches longer than the other piece. (2) The length of the shorter piece is 1/3 the length of the longer piece. (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: 比較長的那一段長度=? (1)即L-S=20時，所以無法判定，insufficient。 (2)即S=(1/3)L時，所以無法判定，insufficient。 (3)由(1)(2)合併，L- (1/3)L=20，L=30，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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154. The sum of positive integers x and y is 77. What is the value of xy? (1) x = y + 1 (2) x and y have the same 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】D 【思路】題目問: xy=? 已知x+y=77 (1) 即x=y+1時 => x=39, y=38，所以可以判定，sufficient。 (2)即x, y有相同十位數時 只有(38, 39)可以符合，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------155. Did it take Pei more than 2 hours to walk a distance of 10 miles along a certain trail? (1 mile = 1.6 kilometers, rounded to the nearest tenth.) (1) Pei walked this distance at an average rate of less than 6.4 kilometers per hour. (2)

On average, it took Pei more than 9 minutes per kilometer to walk this 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】D 【思路】題目問: 花超過兩個小時走完10英哩? 注意題目敘述，10英哩=16公里 速度16/2=8(km/hr) (1)即v<6.4(km/hr)時 所以可以判定，sufficient。 (2)即1公里花超過9分鐘時 v= 1/(9/60)= 60/9=6.6(km/hr) 所以可以判定，sufficient --------------------------------------------------------------------------------------------------------------------------------------------------------

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156. A store sells milk in cartons of two sizes, one-half gallon and one-quarter gallon. If the store sold a total of 300 cartons of milk yesterday, how many gallons of milk did it sell yesterday? (1) Yesterday the store sold 120 one-quarter-gallon cartons of milk. (2)

Yesterday the store sold 90 gallons of milk in one-half-gallon cartons.

(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: 昨天賣了多少加侖? 已知x+y=300，問(1/2)x+(1/4)y=? (1)即y=120時 => x=180，所以可以判定，sufficient。 (2)即(1/2)x =90時 => x=180，y=120，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------157. If n is a positive integer and r is the remainder when 4 + 7n is divided by 3, what is the value of r? (1) n + 1 is divisible by 3. (2) n > 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】A 【思路】題目問: r=? 已知4+7n=3a+r (1) 即n+1=3b時 4+7n = 4+7(3b-1) = 21b-3 =3(7b-1) => r=0，所以可以判定，sufficient。 (2) 即n>20時，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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158. Each of the 25 balls in a certain box is either red, blue, or white and has a number from 1 to 10 painted on it. If one ball is to be selected at random from the box, what is the probability that the ball selected will either be white or have an even number painted on it? (1) The probability that the ball will both be white and have an even number painted on it is 0. (2) The probability that the ball will be white minus the probability that the ball will have an even number painted on it is 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】E 【思路】題目問: 選出的球是白色或者為偶數的機率是多少? 即P(W∪E)= P(W)+P(E)- P(W∩E)=? (1)即P(W∩E)=0時，所以無法判定，insufficient。 (2)即P(W)-P(E)=0.2時，所以無法判定，insufficient。 (3)由(1)(2)合併，還是無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------159. 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) n is not divisible by 2. (2) n is not 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(n - 1)(n + 1)除以24的餘數? 直接帶入數字判斷 (1) n=1時，=> (n - 1)(n + 1) 餘數0； n=3時，=> (n - 1)(n + 1) 餘數8； n=5時，=> (n - 1)(n + 1) 餘數0； 所以無法判定，insufficient。 (2) n=1時，=> (n - 1)(n + 1) 餘數0； n=2時，=> (n - 1)(n + 1) 餘數3； n=4時，=> (n - 1)(n + 1) 餘數15； n=5時，=> (n - 1)(n + 1) 餘數0； 所以無法判定，insufficient。 (3) 由(1)(2)合併 n=1時，=> (n - 1)(n + 1) 餘數0； n=5時，=> (n - 1)(n + 1) 餘數0； n=7時，=> (n - 1)(n + 1) 餘數0； n=11時，=> (n - 1)(n + 1) 餘數0； 所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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160. If Company M ordered a total of 50 computers and printers and Company N ordered a total of 60 computers and printers, how many computers did Company M order? (1) Company M and Company N ordered the same number of computers. (2) Company N ordered 10 more printers than Company M. (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: M公司訂了多少電腦? 已知C1+P1=50，C2+P2=60，問C1=? (1)即C1=C2時，所以無法判定，insufficient。 (2)即P2-P1=10時，只知道C2-C1=0，所以無法判定，insufficient。 (3)由(1)(2)合併，還是無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------161. If the average (arithmetic mean) of the five numbers x, 7, 2, 16, and 11 is equal to the median of the five numbers, what is the value of x? (1) 7 < x < 11 (2) x is the median of the five 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】D 【思路】題目問: x=? 已知算術平均數=中位數 由小到大排列，2, 7, 11, 16，再討論x (1)即7<x<11時，(2+ 7+11+16+x)/5=x，=> x=9，所以可以判定，sufficient。 (2)即x是中位數時，即7≦x≦11時，(2+ 7+11+16+x)/5=x，=> x=9，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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162. 錯 What fraction of this year's graduating students at a certain college are males? (1) Of this year's graduating students, 33 percent of the males and 20 percent of the females transferred from another college. (2) Of this year's graduating students, 25 percent transferred from another 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】C 【思路】題目問: 畢業生中男生所佔的比例? 求M/(M+F) =? (1) 即0.33M+0.2F=transfer時 所以無法判定，insufficient。 (2) 即0.25(M+F)=transfer時 所以無法判定，insufficient。 (3)由(1)(2)合併，0.08M=0.05F，M:F=5:8，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------163. Linda put an amount of money into each of two new investments, A and B, that pay simple annual interest. If the annual interest rate of investment B is 1.5 times that of investment A, what amount did Linda put into investment A? (1) The interest for 1 year is \$50 for investment A and \$150 for investment B. (2) The amount that Linda put into investment B is twice the amount that she put into investment 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】E 【思路】題目問: 多少錢投資在A上? 即A: x(1+n*Ra) B: y(1+n*Rb)，又Rb=1.5Ra，求x=? (1) 即x*Ra=50且y*1.5Ra=150時 => x/y =1/2，所以無法判定，insufficient。 (2)即y=2x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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164. 錯 At a two-day seminar, 90 percent of those registered attended the seminar on the first day. What percent of those registered did not attend the seminar on either day? (1) A total of 1,000 people registered for the two-day seminar. (2) Of those registered, 80 percent attended the seminar on the second day. (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: 多少登記的人沒參加另一天研討會的比例? A是第一天參加，B是第二天參加。P(A)=0.9，求P(A-B)+P(B-A)=? 考Venn Diagram關於Overlapping觀念。

P(A-B)+P(B-A)=[P(A)-P(A∩B)]+[P(B)-P(A∩B)] (1)即A∪B=1,000時 不知道P(B)和P(A∩B)，所以無法判定，insufficient。 (2)即P(B)=0.8時 不知道P(A∩B)，所以無法判定，insufficient。 (3)由(1)(2)合併，不知道P(A∩B)，所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------165. Of the 1,400 college teachers surveyed, 42 percent said that they considered engaging in research an essential goal. How many of the college teachers surveyed were women? (1) In the survey, 36 percent of the men and 50 percent of the women said that they considered engaging in research an essential goal. (2) In the survey, 288 men said that they considered engaging in research an essential goal. (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: 多少被訪談的老師是女性? 已知M+F=1,400，問F=? (1)即(0.36M+0.5F) = 0.42*1,400=588時 可以解出F=600，所以可以判定，sufficient。 (2)即288+P*F = 0.42*1,400=588時 不知道比率，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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166. In the xy-coordinate plane, the slope of line l is 3/4. Does line l pass through the point (-2/3, 1/2)? (1) Line l passes through the point (4,4). (2) Line l passes through the point (-4,-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: 直線l會通過(-2/3, 1/2)? 已知直線l斜率=3/4 (1) 即通過(4, 4)時 斜率= 4 - (1/2) / 4 - (-2/3) = 3/4，所以可以判定，sufficient。 (2) 即通過(-4, -2)時 斜率= 4 - (-2) / 4 - (-4) = 3/4，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------167. 錯 If x, y, and z are integers and xy + z is an odd integer, is x an even integer? (1) xy + xz is an even integer. (2) y + xz 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: x是偶數? 已知xy+z是奇數，必為兩種情形 (I)(xy, z)=(奇,偶)或(II) (xy, z)= (偶,奇) (1) xy+xz=x(y+z)是偶數時: =>必為x是偶數或y+z是偶數 (i)x是偶數: 所以可以判定。 (ii)y+z是偶數: 考慮條件(I): z為偶數=> y是偶數，所以x必為奇數。 考慮條件(II): z為奇數=> y是奇數，所以x必為偶數。 所以可以判定。 由(i)(ii)可以知道sufficient。 (2) y+xz=是奇數時: =>必為y是偶數或xz是偶數 (i)y是偶數: 考慮條件(I):出現矛盾，不成立。 考慮條件(II): x可以為奇數，也可以為偶數。 所以無法判定。 (ii)xz是偶數: 考慮條件(I) z為偶數: x可以為奇數，也可以為偶數。 考慮條件(II) z為奇數: 所以x必為偶數。 所以無法判定。 由(i)(ii)可以知道insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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168. On the number line, the distance between x and y is greater than the distance between x and z. Does z lie between x and y on the number line? (1) xyz < 0 (2) xy < 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: z在x和y之間? 直接畫出數線表示最快 y z x z y ----|----------|---|---|----------|--(1) 即xyz<0時 (x,y,z)可能為(+,-,+) (+,+,-)(-,+,+)(-,-,-) 正負符號沒有影響，所以無法判定，insufficient。 (2) 即xy<0時(x,y)可能為(+,-)(-,+) 正負符號沒有影響，所以無法判定，insufficient。 (3)由(1)(2)合併，還是無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------169. If the integers a and n are greater than 1 and the product of the first 8 positive integers is a multiple of an, what is the value of a? (1) an = 64 (2) n = 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: a=? 8! = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 =27 * 32 * 5 * 7 = m * an (1) 即an =64時 an 可能為26, 43,或82，所以無法判定，insufficient。 (2) 即n=6時，因為a>1，所以a只能為2，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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170. 錯 If n and t are positive integers, what is the greatest prime factor of the product nt? (1) The greatest common factor of n and t is 5. (2) The least common multiple of n and t is 105. (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: nt的最大質因數=? 設n=dh, t=dk，其中(h,k)=1，[n,t]=dhk (1)(n,t)=5時，所以n=5a，t=5b，其中(a,b)=1， 可知nt=52 * a * b，其中a, b也可能為最大質因數 所以無法判定，insufficient。 (2)[n,t]=105=3*5*7=dhk時， 因為nt=d2 *h*k = 105d = 3*5*7*d， 所以nt的最大質因數是7，可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------171. Is | x - y | > |x | - |y |? (1) y < x (2) xy < 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】B 【思路】題目問: |x - y| > |x|- |y|? 直接代數字去判斷會比較簡單。 (1) 即y<x時 (i)(x,y)=(5,2)時，不成立， (ii)(x,y)=(5,-2)時，成立， (iii)(x,y)=(-2, 3)時，成立， (iv)(x,y)=(-2, -3)時，不成立， 所以無法判定，insufficient。 (2) 即xy<0時 x和y必為不同號，| x - y |只會越大或越小 => 恆成立，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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172. 錯 If m, k, x, and y are positive numbers, is mx + ky > kx + my? (1) m > k (2) 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】C 【思路】題目問: mx+ky > kx+my? 注意，不知道正負號時，不要亂消去，因為會影響到不等式方向。 (1) 即m>k時 注意(m-k)x > (m-k)y，所以m-k可以消去，也就是判斷 x>y? 不知道x, y關係，所以無法判定，insufficient。 (2) 即x>y時 注意(x-y)m > (x-y)k，所以x-y可以消去，也就是判斷 m>k? 不知道m, k關係，所以無法判定，insufficient。 (3)由(1)(2)合併，恆成立，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------173. If N is a positive integer, is the units digit of N equal to zero? (1) 14 and 35 are factors of N. (2) N = (25)(32)(57)(76) (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的個位數=0? (1)14和35是N的因數時: N=14a=35b =[14,35]k=2*5*7*k =10*7k => 個位數必為0，所以可以判定，sufficient。 (2)即N=(25)(32)(57)(76)時 => N = 10*….. => 個位數必為0，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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174. In the xy-plane, what is the y-intercept of line l? (1) The slope of line l is 3 times its y-intercept. (2) The x-intercept of line l is -1/3. (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: 直線l的y截距? 直線y=ax+b，求b=? (1)即a=3b時 => y=3bx+b，不知道b，所以無法判定，insufficient。 (2)即(-1/3,0)代入=> (-1/3)a+b=0時 => b=(1/3)a，但不知道a，所以無法判定，insufficient。 (3)由(1)(2)合併，還是無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------175. The sum of the integers in list S is the same as the sum of the integers in list T. Does S contain more integers than T ? (1) The average (arithmetic mean) of the integers in S is less than the average of the integers in T. (2) The median of the integers in S is greater than the median of the integers in 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】A 【思路】題目問: 集合S比集合T有更多整數? 已知Sum(S)=Sum(T) (1) 即Mean(S)<Mean(T)時 => Num(S)>Num(T)，所以可以判定，sufficient。 (2)中位數的大小並不會影響樣本數，所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------176. If x and y are positive integers, is the product xy even? (1) 5x - 4y is even. (2) 6x + 7y 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】D 【思路】題目問: xy是偶數? (1) 即5x-4y是偶數時 => x必為偶數，所以可以判定，sufficient。 (2)即6x+7y是偶數時 => y必為偶數，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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177. If x and y are positive integers, what is the value of xy? (1) The greatest common factor of x and y is 10. (2) The least common multiple of x and y is 180. (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=? (1)即(x,y)=10時，xy=102*h*k，所以無法判定，insufficient。 (2)即[x,y]=180=dhk時，xy=d2*h*k=180d，所以無法判定，insufficient。 (3)由(1)(2)合併，xy=(x,y)* [x,y]=1800，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------178. If there are more than two numbers in a certain list, is each of the numbers in the list equal to 0? (1) The product of any two numbers in the list is equal to 0. (2) The sum of any two numbers in the list is equal to 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】B 【思路】題目問: 數列的數字都是0嗎? (1) 即任兩數相乘為0時 => 只要其中有一數為0，乘積就是0 Ex: {0, 1, 0},{0, 0, 0}，所以並不能代表整個數列均為0 所以無法判定，insufficient。 (2) 即任兩數相加總和為0時 必為整個數列均為0，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

86

179. A manufacturer conducted a survey to determine how many people buy products P and Q. What fraction of the people surveyed said that they buy neither product P nor product Q? (1) 1/3 of the people surveyed said that they buy product P but not product Q. (2) 1/2 of the people surveyed said that they buy product Q. (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: 既不買P也不買Q的人所佔的比例? 即P(A’∩B’)= P(A∪B)’=1-P(A∪B)=1-[ P(A)+P(B)- P(A∩B)]? (1)即P(A-B)=1/3時，P(A)- P(A∩B)=1/3，不知道P(B)，所以無法判定，insufficient。 (2)即P(B)=1/2時，不知道P(A)和 P(A∩B)，所以無法判定，insufficient。 (3)由(1)(2)合併，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------180. 錯 Are x and y both positive? (1) 2x – 2y = 1 (2) x/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】C 【思路】題目問: x, y皆為正? (1) 即x - y=1/2時 (x, y)=(+,+)(+,-)(-,-)都可以成立 所以無法判定，insufficient。 (2) 即x/y > 1時 (x, y)=(+,+)(-,-)都可以成立 所以無法判定，insufficient。 (3)由(1)(2)合併，x=y+(1/2)代入不等式 => 1+(1/y)>1 => 1/y>0 =>y>0 => x必為正，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

87

181.

In the figure shown, what is the value of x? (1) The length of line segment QR is equal to the length of line segment RS. (2) The length of line segment ST is equal to the length of line segment TU. (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=? 直接在圖形上標記 (1) 即QR=RS時，角RQS=角RSQ=m，所以無法判定，insufficient。 (2) 即ST=TU時，角TSU=角TUS=n，所以無法判定，insufficient。 (3) 由(1)(2)合併，角QRS+角STU+90=180 => (180-2m)+(180-2n)=90， => m+n=135 又角RSQ + x +角TSU =180 => x= 180 - (m+n) = 45，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------182. In a certain year, the difference between Mary's and Jim's annual salaries was twice the difference between Mary's and Kate's annual salaries. If Mary's annual salary was the highest of the 3 people, what was the average (arithmetic mean) annual salary of the 3 people that year? (1) Jim's annual salary was \$30,000 that year. (2) Kate's annual salary was \$40,000 that 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】B 【思路】題目問: 三個人薪水的算術平均? 已知M-J=2(M-K)=>M+J=2K，問(M+J+K)/3=? (1) 即J=30,000時，不知道K，所以無法判定，insufficient。 (2) 即K=40,000時，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

88

183. The positive integers x, y, and z are such that x is a factor of y and y is a factor of z.

Is z even?

(1) x is even. (2) y 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】D 【思路】題目問: z是偶數? 已知x|y，y|z (1) 即x是偶數時，=> y是偶數 => z是偶數，所以可以判定，sufficient。 (2) 即y是偶數時，=> z是偶數，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------184. Jack and Mark both received hourly wage increases of 6 percent. After the wage increases, Jack's hourly wage was how many dollars per hour more than Mark's? (1) Before the wage increases, Jack's hourly wage was \$5.00 per hour more than Mark's. (2) Before the wage increases, the ratio of Jack's hourly wage to Mark's hourly wage was 4 to 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】A 【思路】題目問: J每小時薪水比M薪水多了多少? 即1.06J - 1.06M=? (1) 即J-M=5時，所以可以判定，sufficient。 (2) 即J:M=4:3時，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

89

185. Is x between 0 and 1? (1) x is between -1/2 and 3/2 (2) 3/4 is 1/4 more than 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】B 【思路】題目問: 0<x<1 ? (1) 即-1/2< x< 3/2時，所以無法判定，insufficient。 (2) 即3/4 – x =1/4時，=> x=1/2，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------186. If 0 < x < y, what is the value of (x+y)2/(x-y)2 ? (1) x2 + y2 = 3xy (2) xy = 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】A 【思路】題目問: (x+y)2/(x-y)2=? 即(x2+y2+2xy)/ (x2+y2-2xy)=? (1) 即x2+y2=3xy時，所求=5，所以可以判定，sufficient。 (2) 即xy=3時，不知道x, y的值，所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------187. If x, y, and z are integers greater than 1, what is the value of x + y + z? (1) xyz = 70 (2) x/yz = 7/10 (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】A 【思路】題目問: x+y+z=? 已知x, y, z >1 (1) 即xyz=70=2*5*7時，所求=2+5+7=14，所以可以判定，sufficient。 (2) 即x/yz = 7/10時，=>10x=7yz，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

90

188. Is p + pz = p? (1) p = 0 (2) 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: p + pz = p? 即p(1+z)=p? (1) 即p=0時，恆成立，所以可以判定，sufficient。 (2) 即z=0時，恆成立，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------189. If the operation △ is one of the four arithmetic operations addition, subtraction, multiplication, and division, is (6 △ 2) △ 4 = 6 △ (2 △ 4) ? (1) 3 △ 2 > 3 (2) 3 △ 1 = 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】A 【思路】題目問: (6△2)△4 = 6△(2△4) ? (1) 即3△2 > 3時 △可為+或*，恆成立 所以可以判定，sufficient。 (2) 即3△1 = 3時 △可為*或÷ ，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

91

190.

On the number line shown, is zero halfway between r and s? (1) s is to the right of zero. (2) The distance between t and r is the same as the distance between t and -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: 0位於r和s的中間? 直接在數線上表示最快 (1) 即s在0的右方時 0可能在r的左邊和右邊 所以無法判定，insufficient。 (2) 即t和r的距離= t和-s的距離時，-s可能(i)等於r或(ii)在t的右方 (i) (-s) <-------r-----0-----s------t------------------------->，可以成立。 (ii)<-------r-----------s--0--t--------0--------(-s)---->，無法成立。 所以無法判定，insufficient。 (3)由(1)(2)合併，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------191. If the product of the three digits of the positive integer k is 14, what is the value of k? (1) k is an odd integer. (2) k < 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】E 【思路】題目問: k=? 已知xyz=14=2*7，x, y, z為0~9，求k=100x+10y+z=? (1) 即k是奇數時 =>z必為奇數，z=1或7， ex: {271, 721, 127, 217}，所以無法判定，insufficient。 (2) 即k<700時 (x,y,z)={1,2,7}，ex: {127, 172, 271, 217}，所以無法判定，insufficient。 (3)由(1)(2)合併，ex: {127, 217, 271}，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

92

192. 錯 Of the 75 houses in a certain community, 48 have a patio. How many of the houses in the community have a swimming pool? (1) 38 of the houses in the community have a patio but do not have a swimming pool. (2) The number of houses in the community that have a patio and a swimming pool is equal to the number of houses in the community that have neither a swimming pool nor a patio. (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: 多少房子有游泳池，即a+b=? Patio ~patio

Swim a b

~ Swim c d

48 27 Total:75

a+b+c+d=75，a+c=48，b+d=27 (1)即c=38時 a= 48 - 38 = 10，但是不知道b，所以無法判定，insufficient。 (2)即a=d時 a+b = d+b = 27，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------193. The attendees at a certain convention purchased a total of 15,000 books. How many of these attendees were female? (1) There was a total of 4,000 attendees at the convention. (2) The male attendees purchased an average (arithmetic mean) of 3 books each, and the female attendees purchased an average of 5 books 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】C 【思路】題目問: 多少參加者是女性? 已知x*M+y*F=15,000，問F=? (1) 即M+F=4,000時，所以無法判定，insufficient。 (2) 即3M+5F=15,000時，所以無法判定，insufficient。 (3)由(1)(2)合併，F=1,500，M=2,500，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

93

194. If x and y are positive integers, what is the value of x + y? (1) 2x 3y = 72 (2) 2x 2y = 32 (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=? (1) 即2x 3y = 72=23 * 32 時 => x=3, y=2，所以可以判定，sufficient。 (2) 即2x 2y = 32=25 時 => x+y=5，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------195. What is the value of h? (1) h2 = 36 (2) h2 + 12h = -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】B 【思路】題目問: h=? (1) 即h2 = 36時，h=6或-6，所以無法判定，insufficient。 (2) 即h2 + 12h = -36時，即(h+6)2=0，h=-6，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------196. How many odd integers are greater than the integer x and less than the integer y? (1) There are 12 even integers greater than x and less than y. (2) There are 24 integers greater than x and less than 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: 多少個奇數大於x且小於y? 即x<多少個奇數<y (1) 即x< 12個偶數 <y時 不確定有多少數介於x, y，可能比偶數個數多1，少1，或等於 所以無法判定，insufficient。 (2) 即x<24個整數<y時 因為奇偶的對偶性，可知有一半是奇數，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

94

197. 錯 Jason's salary and Karen's salary were each p percent greater in 1998 than in 1995. What is the value of p? (1) In 1995 Karen's salary was \$2,000 greater than Jason's. (2) In 1998 Karen's salary was \$2,440 greater than Jason'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: p=? 設1995: Janson's salary = J Karen's salary = K 1998: Janson's salary = J (1+p) Karen's salary = K (1+p) (1) 即K – J =2,000時 所以無法判定，insufficient。 (2) 即K (1+p) - J (1+p)=2,440時 => K – J +Kp - Jp=2,440 => (K - J)(1+p) =2,440 所以無法判定，insufficient。 (3) 由(1)(2)合併，1+p=2,440/2,000=1.22 P=22%，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------198. 錯 Did one of the 3 members of a certain team sell at least 2 raffle tickets yesterday? (1) The 3 members sold a total of 6 raffle tickets yesterday. (2) No 2 of the members sold the same number of raffle tickets yesterday. (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: 至少有一個人賣兩張票? (1)即a+b+c=6時，平均值=2，所以可以判定，sufficient。 (2)即a, b, c相異時，(a,b,c)=(0,1,2) (1,2,3) (3,4,5)，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

95

199. If s and t are two different numbers on the number line, is s + t equal to 0? (1) The distance between s and 0 is the same as the distance between t and 0. (2) 0 is between s and 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】A 【思路】題目問: s+t=0? 已知s≠t (1) 即|s|=|t|時 即兩點不重疊且與0的距離相等 => s+t=0，所以可以判定，sufficient。 (2) 即0在s和t之間時 所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------200. Of the 1,000 companies responding to a certain survey, what percent indicated that they had a business recovery plan? (1) 200 of the companies did not indicate that they had a business recovery plan. (2) The number of companies that indicated that they had a business recovery plan was 4 times the number that did not indicate that they had a business recovery plan. (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: 在調查中公司有備份計畫的比例是多少? (1)即有200家公司沒有企業備份計畫時 所求=800/1,000=80%，所以可以判定，sufficient。 (2)即P=4P’時 又P+P’=1,000=> P’=200，P=800 所求=800/1,000=80%，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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201. Martha bought an armchair and a coffee table at an auction and sold both items at her store. Her gross profit from the purchase and sale of the armchair was what percent greater than her gross profit from the purchase and sale of the coffee table? (1) Martha paid 10 percent more for the armchair than for the coffee table. (2) Martha sold the armchair for 20 percent more than she sold the coffee 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】E 【思路】題目問: 扶椅的淨收入大於咖啡桌淨收入的比例? Gross Profit=Sale - Purchase 問(SA-PA)/(SC-PC)=? (1)即PA=1.1PC時，所以無法判定，insufficient。 (2)即SA=1.2SC時，所以無法判定，insufficient。 (3)由(1)(2)合併，(SA-PA)/(SC-PC)=(1.2SC-1.1PC)/(SC-PC)，所以無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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202. 錯 Each employee of Company Z is an employee of either Division X or Division Y, but not both. If each division has some part-time employees, is the ratio of the number of full-time employees to the number of part-time employees greater for Division X than for Company Z? (1) The ratio of the number of full-time employees to the number of part-time employees is less for Division Y than for Company Z. (2) More than half of the full-time employees of Company Z are employees of Division X, and more than half of the part-time employees of Company Z are employees of Division 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】D 【思路】題目問: Division X的全職員工:兼職員工的比例大於Company Z? 其中f(z)=f(x)+f(y) ; p(z)=p(x)+p(y) 問f(x)/p(x) > f(z)/p(z) ? (1)即f(y)/p(y) < f(z)/p(z)時: 將y置換掉。 => [f(z) – f(x)]/[p(z) – p(x)]< f(z)/p(z) => p(z) [f(z) – f(x)] < f(z) [p(z) – p(x)] => f(z) p(z) - f(x) p(z) < f(z)p(z) - f(z)p(x) => f(z) p(x) < f(x)p(z) => f(x)/p(x) > f(z)/p(z) 所以可以判定，sufficient。 (2)即f(x)> (1/2)f(z)，p(y)> (1/2)p(z)時 =>2f(x)>f(z)，2p(y)>p(z) 將y置換掉。 =>2[p(z)-p(x)]>p(z) =>p(z)>2p(x) 又2f(x)>f(z) 所以p(z) 2f(x) > 2p(x) f(z) => p(z) f(x) > p(x) f(z) => f(x)/p(x) > f(z)/p(z) 所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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203. What is the remainder when the positive integer x is divided by 6? (1) When x is divided by 2, the remainder is 1; and when x is divided by 3, the remainder is 0. (2) When x is divided by 12, 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】D 【思路】題目問: 正整數x除以6的餘數? (1)即x=2a+1=3b時 =>x=6m+3，餘數=3，所以可以判定，sufficient。 (2)即x=12c+3=6(2c)+3時 =>餘數=3，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------204. 錯 On a certain sight-seeing tour, the ratio of the number of women to the number of children was 5 to 2. What was the number of men on the sight-seeing tour? (1) On the sight-seeing tour, the ratio of the number of children to the number of men was 5 to 11. (2) The number of women on the sight-seeing tour was less than 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】C 【思路】題目問: 觀光旅行中多少是男人? 已知W:C=5:2 (1)即C:M=5:11時 只知比例，不知道人數，所以無法判定，insufficient。 (2)即W<30時 不知道M，所以無法判定，insufficient。 (3)由(1)(2)合併，W:M:C=25:22:10，又25r<30，=>r=1，M=22，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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205. Ellen can purchase a certain computer at a local store at the price of p dollars and pay a 6 percent sales tax. Alternatively, Ellen can purchase the same computer from a catalog for a total of q dollars, including all taxes and shipping costs. Will it cost more for Ellen to purchase the computer from the local store than from the catalog? (1) q - p < 50 (2) q = 1,150 (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: 1.06p > q ? (1) 即q-p<50時，q<p+50，所以無法判定，insufficient。 (2) 即q=1,150時，不知道p，所以無法判定，insufficient。 (3)由(1)(2)合併，p>1,100 => 1.06p>1,166會大於q，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------206. If p is a positive integer, what is the value of p? (1) p is a prime number. (2) p 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: p=? (1) 即p是質數時，p=2, 3, 5, 7，所以無法判定，insufficient。 (2) 即p=3a時，p=3, 6, 9，所以無法判定，insufficient。 (3) 由(1)(2)合併，p=3，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------207. Is y < 2x? (1) y/4 < x/2 (2) (y-2x)/3 < 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: y < 2x? (1) 即y/4 < x/2時，=> y < 2x，所以可以判定，sufficient。 (2) 即(y-2x)/3 < 0時，=> (y-2x)<0 => y < 2x，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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208. Is x > y ? (1) x + y < 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】B 【思路】題目問: x > y? (1) 即x + y < 0時 => (x, y)可為(+,-)(-,+)(-,-)，所以無法判定，insufficient。 (2) 即x - y > 0時 => x > y，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------209. For each month of next year, Company R's monthly revenue target is x dollars greater than its monthly revenue target for the preceding month. What is Company R's revenue target for March of next year? (1) Company R's revenue target for December of next year is \$310,000. (2) Company R's revenue target for September of next year is \$30,000 greater than its revenue target for June of next 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】C 【思路】題目問: 明年三月的營收目標是多少? (1)即明年十二月的營收是310,000時 所求=310,000-9x，不知道x，所以無法判定，insufficient。 (2)即明年九月的營收比明年六月的營收多30,000時 所以無法判定，insufficient。 (3)由(1)(2)合併， 3x=30,000，x=10,000 所求=310,000-9x=220,000，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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210. 錯 Is m + z > 0? (1) m - 3z > 0 (2) 4z - 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】C 【思路】題目問: m + z > 0? (1)即m - 3z > 0時 m > 3z，不知道z的正負，所以無法判定，insufficient。 (2)即4z - m > 0時 m < 4z，不知道z的正負，所以無法判定，insufficient。 (3)由(1)(2)合併，3z < m < 4z，=> z >0 => m >0，所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------211. If k is a positive integer, then 20k is divisible by how many different positive integers? (1) k is prime. (2) k = 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】B 【思路】題目問: 20k有多少個相異正因數? 已知20k=22*5*k (1) 即k是質數時，k可能為2, 5或其他質數 (i)k=2，(3+1)(1+1)=8 (ii)k=5，(2+1)(2+1)=9 (iii)k=其他質數，(2+1)(1+1) (1+1)=12 所以無法判定，insufficient。 (2) 即k=7時 (2+1)(1+1) (1+1)=12，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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212. This morning, a certain sugar container was full. Since then some of the sugar from this container was used to make cookies. If no other sugar was removed from or added to the container, by what percent did the amount of sugar in the container decrease? (1) The amount of sugar in the container after making the cookies would need to be increased by 30 percent to fill the container. (2) Six cups of sugar from the container were used to make the cookies. (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: 糖減少多少比例? 已知1 – x =剩下來的糖，問x=? (1)即x=30%時，所以可以判定，sufficient。 (2)即用掉6杯的糖製造餅乾時，不知道全部的量，所以無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------213. If a and b are positive numbers, what are the coordinates of the midpoint of line segment CD in the xy-plane? (1) The coordinates of C are (a, 1 - b). (2) The coordinates of D are (1 - 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: 線段CD的中點座標? (1) 即C (a, 1-b)時，不知道D座標，所以無法判定，insufficient。 (2) 即D (1- a, b)時，不知道C座標，所以無法判定，insufficient。 (3) 由(1)(2)合併，所求=(1/2, 1/2)，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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214. The cost of a square slab is proportional to its thickness and also proportional to the square of its length. What is the cost of a square slab that is 3 meters long and 0.1 meter thick? (1) The cost of a square slab that is 2 meters long and 0.2 meter thick is \$160 more than the cost of a square slab that is 2 meters long and 0.1 meter thick. (2) The cost of a square slab that is 3 meters long and 0.1 meter thick is \$200 more than the cost of a square slab that is 2 meters long and 0.1 meter thick. (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: 長3公尺厚0.1公尺的正方型木板成本? 已知成本和厚度成正比，也和長度平方成正比 (1)即長2公尺厚0.2公尺的正方型木板 – 長2公尺厚0.1公尺正方型的木板 = 160時 => 22*0.2*r - 22*0.1*r = 160 => r=400 所求=32*0.1*r = 360，所以可以判定，sufficient。 (2)即長3公尺厚0.1公尺的正方型木板 – 長2公尺厚0.1公尺正方型的木板 = 200時 => 32*0.1*r - 22*0.1*r = 200 => r=400 所求=32*0.1*r = 360，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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215. 錯 If Bob produces 36 or fewer items in a week, he is paid x dollars per item. If Bob produces more than 36 items in a week, he is paid x dollars per item for the first 36 items and 1.5 times that amount for each additional item. How many items did Bob produce last week? (1) Last week Bob was paid a total of \$480 for the items that he produced that week. (2) This week Bob produced 2 items more than last week and was paid a total of \$510 for the items that he produced this week. (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: 上禮拜製造了多少物品? n≦36時 => nx n>36時 => 36x+(n-36)*1.5x 求n=? (1) n ≦ 36時 => nx = 480 n > 36時 => 36x + 1.5x(n-36) = 480 所以無法判定，insufficient。 (2) n ≦ 36時 => (n+2)x=510 n > 36時 => 36x+ (n+2 - 36)*1.5x=510 無法判定，insufficient。 (3)由(1)(2)合併 (i) n ≦ 36時 => nx = 480，(n+2)x=510 => 2x=30 => x=15，n=32 (ii) n > 36時 => 36x + 1.5x(n-36) = 480 36x+ (n-34)*1.5x =510 => 3x=30 => x=10，n=44 所以n有兩個值，無法判定，insufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------216. Is x < y? (1) 2x < 3y (2) xy > 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? (1) 即x < (3/2)y時 x可能介於y和(3/2)y 所以無法判定，insufficient。 (2) 即xy < 0時，只知道(x,y)=(+,+)(-,-)，所以無法判定，insufficient。 (3)由(1)(2)合併，還是無法判定，insufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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217. 錯 a1 , a2 , a3 , . . . , a15 In the sequence shown, an = an - 1 + k, where 2≦n≦15 and k is a nonzero constant. How many of the terms in the sequence are greater than 10? (1) a1 = 24 (2) a8 = 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】B 【思路】題目問: 序列中多少項大於10? 也就是公差為k的等差數列，其中k是非0常數 (1) 即a1=24時 a2= a1+k=24+k，a3= a2+k=24+2k 不知道k值，所以無法判定，insufficient。 (2) 即a8=10時，即中位數為10 (i)k>0，那麼a9~a15有7個大於10。 (ii)k<0，那麼a1~a7有7個大於10。 所以可以判定，sufficient。 -------------------------------------------------------------------------------------------------------------------------------------------------------218. Is 4z > -6? (1) z < 7 (2) z > -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: 4z > -6? 即z > -(3/2)? (1) 即z < 7時，所以無法判定，insufficient。 (2) 即z > -1時，所以可以判定，sufficient。 --------------------------------------------------------------------------------------------------------------------------------------------------------

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ds1 solution

ds1 solution