IL Foundation Grade 7 - Math by infinity-learn

IL FOUNDATION SERIES

MATHEMATICS

IL Foundation Series - Mathematics Class 7

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ISBN 978-81-985304-4-8

Second Edition

1.1 INTRODUCTION

Natural numbers: These are the collection of numbers which are countable.

Examples: 1, 2, 3, 4, ..., etc.

Whole numbers: These are the collection of natural numbers along with zero.

Examples: 0, 1, 2, 3, 4, ..., etc.

Integers: These are the collection of whole numbers along with negative numbers.

Examples: –14, –3, 0, 2, 7, 19, etc.

Integers include positive numbers, which are greater than zero; negative numbers which are less than zero; and zero which is neither a negative number nor a positive number.

1.2 BASICS OF INTEGERS

1.2.1

Integers on a number line

A number line helps compare numbers placed at equal intervals on an infinite line that goes both ways. Integers, like other numbers, can also be shown on a number line.

INTEGERS

On a number line, positive numbers are to the right of zero, and negative numbers are to the left. The farther left we go, the bigger the numbers get, while the farther right we go, the smaller they become.

1.2.2 Addition and subtraction of integers

Addition of integers

Adding integers means finding the total when you combine two or more integers. The sum might increase or decrease depending on whether the integers are positive or negative.

Rules for adding integers

• If both integers have the same signs, add their absolute values and keep the sign the same.

• If one integer is positive and the other is negative, subtract their absolute values and use the sign of the larger number.

Example: Add the integers:

i. (–2) + (–8)

Solution:

i. (–2) + (–8) = –10

Additive inverse

ii. (–2) + (8)

ii. (–2) + (8) = 6

The additive inverse of a number is another number that, when added to the original number, results in zero.

In simpler terms, if you have a number a, its additive inverse is denoted as –a, and when you add a and –a together, you get zero.

Example: The additive inverse of 7 is –7, and the additive inverse of –7 is 7.

Subtraction of integers

Subtracting integers means finding the difference between two integers. The result might increase or decrease depending on whether the integers are positive or negative.

Rules for subtracting integers

• Convert the subtraction into an addition problem by using the opposite (additive inverse) of the number being subtracted.

• Then, follow the rules for adding integers and solve the resulting addition problem.

Example: Subtract the integers: (–1) – (–2)

Solution:

(–1)–(–2) = –1 + 2 = 1

1.3 PROPERTIES OF ADDITION AND SUBTRACTION OF INTEGERS

1.3.1 Closure under addition

For any two integers a and b, (a+b) is an integer.

When we add two integers, their sum is also an integer.

Examples: 15 + 12 = 27; 20 + (-11) = 9

1.3.2 Closure under subtraction

For any two integers a and b, (a – b) is an integer.

When we subtract two integers, their difference is also an integer.

Examples: 5 – 10 = – 5; 10 – (–1) = 11

1.3.3 Commutative property

Commutative property of addition

For any two integers a and b, a+b=b+a. So, addition is commutative for integers.

Example: 5 + (–16) = (–16) + 5

Commutative property of subtraction

For any two integers a and b, a – b ≠ b-a. So, subtraction is not commutative for integers.

Example: 11 – 3 ≠ 3 – 11

1.3.4 Associative property

Associative property of addition

For any three integers a, b, and c, (a+b) +c=a+ (b+c). So, addition is associative for integers.

Example: (7 + 20) + 3 = 7 + (20 + 3)

Associative property of subtraction

For any three integers a, b, and c, (a – b) – c ≠ a – (b-c). So, subtraction is not associative for integers.

Example: ( ) ( ) 3213[21] −−−≠−−− 

1.3.5 Additive identity

Zero is an additive identity for integers.

For any integer a, a + 0 = a = 0 + a.

Example: ( ) 202−+=−

SOLVED EXAMPLES

Example 1: At a village, the temperature was -2°C on Thursday, and then it dropped by 4°C on Friday. What was the temperature of the village on Friday? On Saturday, it rose by 6°C. What was the temperature on this day?

Solution:

Given, the temperature on Thursday = -2°C

To find the temperature on Friday, we subtract 4°C from Thursday's temperature:

i.e., -2°C - 4°C = - 6°C

So, the temperature on Friday was - 6°C.

To find the temperature on Saturday, we add 6°C to Friday's temperature:

i.e., - 6°C + 6°C = 0°C

So, the temperature on Saturday was 0°C.

Example 2: Find a pair of negative integers whose sum is -10.

Solution:

A pair of negative integers whose sum is -10 can be: -7 and -3

Because: -7 + (- 3) = -10

Example 3: Write a pair of integers whose sum gives an integer greater than both the integers.

Solution:

Let's consider the pair (5, 8).

Their sum is 5 + 8 = 13, which is greater than both 5 and 8.

Example 4: Fill in the blanks to make the following equations true and mention the property:

i. 3 + (7 + 4) = (3 + 7) +_____

ii. -24 + ______ = -24

iii. (-5) + 12 =_____+ (-5)

Solution:

i. 3 + (7 + 4) = (3 + 7) + 4 [By associative property of addition]

ii. -24 + 0 = -24 [By additive identity]

iii. (-5) + 12 = 12 + (-5) [By commutative property of addition]

1.4 MULTIPLICATION OF INTEGERS

Multiplication of integers is a mathematical operation that combines two integers to find their product. We have learned that when multiplying two positive integers, the product is positive.

For any two positive integers a and b, a×b = a×b.

Example: 11 × 4 = 44

1.4.1 Multiplication of a positive and a negative integer

The product of a positive integer and a negative integer is a negative integer.

For any two positive integers a and b, ( ) ( ) ().×−=−×=−× ababab

Example: ( ) ( ) ( ) 434312 −×=−×=−

1.4.2 Multiplication of two negative integers

The product of two negative integers is a positive integer.

For any two positive integers a and b, ( ) (). −×−=× abab

Example: ( ) ( ) 254100−×−=

1.4.3 Product of three or more negative integers

The product of three negative integers is a negative integer.

Example: ( ) ( ) ( ) 13515 −×−×−=−

The product of four negative integers is a positive integer.

Example: ( ) ( ) ( ) ( ) 125110 −×−×−×−=

It is observed that the product of an odd number of negative integers is a negative integer, and the product of an even number of negative integers is a positive integer.

Note: The product of any number of positive integers is always a positive integer.

1.5 PROPERTIES OF MULTIPLICATION OF INTEGERS

1.5.1 Closure under multiplication

For any two integers a and b, a×b is an integer.

Example: ( ) ( ) 21326−×−=

1.5.2 Commutativity of multiplication

For any two integers a and b, a×b=b×a.

Example: ( ) ( ) 5115 ×−=−×

1.5.3 Multiplication by zero

For any integer a, a × 0 = 0 × a = 0.

Example: ( ) 700−×=

1.5.4 Multiplicative identity

1 is the multiplicative identity for all the integers.

For any integer a, ×=×= 11 aaa

Example: ( ) 144 ×−=−

1.5.5 Associativity for multiplication

For any three integers a, b, and c,,()(). ××=×× cabcabc

Example: ( ) ( ) 243243 × ×−×=×−  

1.5.6 Distributive property

Distributivity of multiplication over addition

For any three integers a, b, and c, ( ) ( ) ( ) . ×+=×+× abcabac

Example: ( ) ( ) 43043[40] ×−+=×−+×  

LHS = ( ) ( ) 4304312 ×−+=×−=−   and RHS ( ) [ ] 434012012 =×−+×=−+=−

Distributivity of multiplication over subtraction

For any three integers a, b, and c, a× ( ) ( ) ( ) −=×−× bcabac

Example: ( ) ( ) ( ) ( ) ( ) 1721712 −×−−=−×−−−×

LHS = ( ) ( ) ( ) 172(1)99 −×−−=−×−=

and RHS = ( ) ( ) ( ) ( ) 171272729 −×−−−×=−−=+=

Example 1: Find each of the following products:

i. ( ) 125(3)−××− ii. ( ) ( ) 123(4) −×−××−

Solution:

i. ( ) ( ) ( ) 12531253 × −××−=−×−   ( ) ( ) 603=−×− =180

ii. ( ) ( ) 123(4) −×−××− ( ) ( ) ( ) 1234 =−×−××−  

(2)(12)=×− -24 =

Example 2: In a quiz with 20 questions, each correct answer earns 5 points, while each incorrect answer results in a deduction of 2 points. If Maya answers all the questions but only gets 15 of them correct, what is her total score?

Solution:

Given, each correct answers earn 5 points. And each incorrect answers result in a deduction of 2 points.

Maya answered 15 questions correctly and 5 questions incorrectly.

Points earned for correct answers: 15 × 5 = 75 points

Points deducted for incorrect answers: 5 × ( ) 210points−=−

Total score = Points earned for correct answers + Points deducted for incorrect answers

Total score = 75 + (-10) = 65 points

So, Maya's total score is 65 points.

Example 3: Verify, ( ) ( ) ( ) 812(2)8128(2). −×+−=−×+−×−

Solution:

LHS ( ) 812(2) =−×+−  ( ) [ ] 8122=−×− ( ) [ ] 810=−× = -80

RHS ( ) ( ) 8128(2) =−×+−×−    9616=−+ -80 =

So, LHS = RHS

Hence verified.

Example 4: During a cooling experiment, a substance needs to be cooled down from 35°C at the rate of 3°C per hour. What will the temperature of the substance be 8 hours into the experiment?

Solution:

Given,

Initial temperature = 35°C

Cooling rate = 3°C per hour

Time = 8 hours

Temperature after 8 hours = Initial temperature - (Cooling rate × Time)

Temperature after 8 hours = 35°C - (3°C/hour × 8 hours)

Temperature after 8 hours = 35°C - 24°C

Temperature after 8 hours = 11°C

So, the temperature of the substance after 8 hours into the experiment will be 11°C.

1.6 DIVISION OF INTEGERS

Division of integers is the process of dividing one integer by another to find out how many times one number can be contained within another.

1.6.1 Division of two positive integers

When dividing a positive integer by another positive integer, a positive quotient is obtained.

Example: 623 ÷=

1.6.2 Division of two negative integers

When dividing a negative integer by another negative integer, a positive quotient is obtained.

Example: ( ) ( ) 623−÷−=

1.6.3 Division of a positive integer by a negative integer

When dividing a positive integer by a negative integer, a negative quotient is obtained.

Example: ( ) 623 ÷−=−

1.6.4 Division of a negative integer by a positive integer

When dividing a negative integer by a positive integer, a negative quotient is obtained.

Example: ( ) 623−÷=−

Note: When dividing integers, the sign of the quotient depends on the signs of the numbers being divided.

1.7 PROPERTIES OF DIVISION OF INTEGERS

1.7.1 Closure property

If a and b are integers, then a ÷ b is not always an integer.

Examples:

i. -28 ÷ 2 = -14 is an integer

ii. -15 ÷ (-5) = 3 is an integer

iii. 7 ÷ 2 = 7 2 is not an integer

1.7.2 Commutative property

If a and b are integers, then a ÷ b ≠ b ÷ a.

So, the commutative property does not hold in the division of integers.

1.7.3 Some other properties

• If a is an integer and a ≠ 0, then a ÷ a = 1.

• If a is an integer, then a ÷ 1 =a.

• If a is an integer and a ≠ 0, then 0 ÷ a = 0, but a ÷ 0 is not defined.

SOLVED EXAMPLES

Example 1: Divide the following:

i. 65 ÷ (-13) ii. (-84) ÷ 12 iii. (-105) ÷ (-21)

Solution:

i. ( ) ( ) 651365135 ÷−=−÷=−

ii. ( ) ( ) 841284127 −÷=−÷=−

iii. ( ) ( ) 10521105215 −÷−=÷=

Example 2: The product of two integers is 234. If one of the integers is –18, determine the other integer.

Solution:

Given that one integer is -18, and the product of the two integers is 234, so to find the other integer:

Other integer = 234 ÷ (-18) = 234 18

= 234 18 = -13

So, the other integer is -13.

Example 3: A bookstore sells novels and magazines. For every novel sold, the store earns a profit of ₹50, while for every magazine, it incurs a loss of ₹10 due to a clearance sale.

i. During a promotion, the store earns a total profit of ₹400 from the sale of 4 novels and some magazines. How many magazines did it sell during this promotion?

ii. In the subsequent month, the store neither gains nor loses money, selling 8 novels. How many magazines did it sell during this period?

Solution:

i. During the promotion:

For every novel sold, the store earns a profit of ₹50.

So, for 4 novels, the profit earned is 4 × 50 = ₹200.

Let's denote the number of magazines sold as x

For every magazine sold, the store incurs a loss of ₹10.

So, for x magazines, the loss incurred is x × (-10) = -10x.

The total profit earned during the promotion is the sum of profits from novels and losses from magazines, which is equal to ₹400. So, we can write the equation:

200 = 400 - 10x

⇒ 10x = 400 -200

⇒ 10x = 200

⇒ x = 200 10

⇒ x = 20

So, the store sold 20 magazines during the promotion.

ii. In the subsequent month:

The store neither gains nor loses money, selling 8 novels.

The profit from selling 8 novels would be 8 × 50 = ₹400.

Since the store neither gains nor loses money, the profit from novels must offset the losses from magazines.

Therefore, the loss incurred from selling magazines must be ₹400.

For every magazine sold, the store incurs a loss of ₹10.

So, the number of magazines sold can be calculated as:

Loss from magazines = -400

⇒ -10x = -400

⇒ x = 400 10

⇒ x = 40

So, the store sold 40 magazines during the subsequent month.

QUICK REVIEW

• Integers are a set of numbers which includes positive, negative, and zero.

• Properties of addition and subtraction of integers:

Integers are closed for addition and subtraction both. That is, a+b and a-b are again integers, where a and b are any integers.

Addition is commutative for integers, i.e., a+b=b+a for all integers a and b

Addition is associative for integers, i.e., (a + b) + c = a + (b + c) for all integers a, b, and c. Integer 0 is the identity under addition. That is, a + 0 = 0 + a = a for every integer a.

• Zero is an additive identity for all the integers.

• The product of a positive and a negative integer is a negative integer, whereas the product of two negative integers is a positive integer.

• The product of an even number of negative integers is positive, whereas the product of an odd number of negative integers is negative.

• Properties under multiplication of integers:

Integers are closed under multiplication. That is, a × b is an integer for any two integers a and b.

Multiplication is commutative for integers. That is, a × b = b × a for any integers a and b The integer 1 is the identity under multiplication, i.e., 1 × a = a × 1 = a for any integer a Multiplication is associative for integers, i.e., (a × b) × c = a × (b × c) for any three integers a, b, and c.

• Under addition and multiplication, integers show a property called distributive property. That is, a × (b + c) = a × b + a × c for any three integers a, b and c.

• Under subtraction and multiplication, integers show a property called distributive property. That is, a × (b - c) = a × b - a × c for any three integers a, b and c.

• 1 is the multiplicative identity for all the integers.

INTEGERS

• When a positive integer is divided by a negative integer, the quotient obtained is negative and vice-versa.

• Division of a negative integer by another negative integer gives positive as quotient.

• Properties of division for integers:

For any non-zero integer a, we have:

a ÷ a = 1

a ÷ 0 is not defined

a ÷ 1 = a

WORKSHEET - 1

1. Evaluate:

i. 13 + (-9)

2. Find the sum of:

i. -103 and -201

3. Find the additive inverse of:

i. 82

4. Subtract:

i. 192 from -76

I. BASICS OF INTEGERS, PROPERTIES OF ADDITION AND SUBTRACTION OF INTEGERS

ii. (-6) + (-24)

ii. -2024 and 1993

ii. -534

ii. -55 from -16

5. Subtract the sum of -726 and 582 from the sum of -500 and 300.

6. The sum of two integers is 105. If one of them is – 53, find the other.

7. Write a pair of integers whose sum gives a negative integer.

8. Simplify: {10 -(-55)} - {(-21+3)}.

9. If a = -3, b =-6, c = 9, verify that, (a+b) + c=a+ (b+c).

10. In a city, the temperature was 8°C on Thursday, and then it dropped by 10°C on Friday. What was the temperature in the city on Friday? On Saturday, it rose by 15°C. What was the temperature on this day?

II. MULTIPLICATION OF INTEGERS, PROPERTIES OF MULTIPLICATION OF INTEGERS

1. Multiply:

i. -14 by -6 ii. 24 by -5

2. Find each of the following products:

i. ( ) 26(4)−××−

3. Find the value of the following expressions:

i. ( ) ( ) 225times −×−×…

ii. ( ) 549 ×−×

ii. ( ) ( ) s 1113 ime 1 t −×−×…

4. What will be the sign of the product if we multiply 15 negative integers and 65 positive integers?

5. What will be the sign of the product if we multiply 111 positive integers and 110 negative integers?

6. Simplify:

i. ( ) 12141219 ×−+×

ii. ( ) ( ) ( ) 674(7) −×−+−×−

7. Add the product of (-6) × 4, and the product of (-6) × (-5).

8. Verify, ( ) ( ) ( ) ( ) ( ) ( ) 115311511(3) −×−−−=−×−−−×−

9. During a refrigeration process, a room needs to be cooled down from 28°C at the rate of 4°C per hour. What will the temperature of the room be 5 hours into the process?

10. In a math exam comprising 25 questions, students receive 3 points for each correct answer and lose 1 point for each incorrect answer. If Rohan answers all the questions but only manages to get 10 correct, what will be his score?

III. DIVISION OF INTEGERS, PROPERTIES OF DIVISION OF INTEGERS

1. Divide: i. 70 ÷ (-14) ii. (-144) ÷ (-12) iii. (-29) ÷ 29

2. Fill in the blanks. i. ( ) 1 41 −÷= ii. 0 24 ÷= iii. 16-2 ÷=

3. What should be divided by (-8) to obtain 16?

4. Find the quotient in each of the following divisions:

i. 728 ÷ (-7) ii. (-234)÷ 13

5. An elevator descends at a rate of 5 metres per minute. If it starts descending from an altitude of 100 metres above ground level, how long will it take to reach an altitude of -300 metres?

6. A scuba diver descends into the ocean at a rate of 12 metres per minute. If the dive begins at sea level, how long will it take for the diver to reach a depth of -180 metres?

7. The product of two integers is -270. If one of them is 18, find the other.

8. Add the product of (-25) and (-7) to the quotient of 154 and (-11).

9. A toy store sells action figures and board games. For every action figure sold, the store earns a profit of ₹8, while for every board game, it incurs a loss of ₹5 due to clearance. During a promotion, the store earns a total profit of ₹66 from the sale of 12 action figures and some board games. How many board games did it sell during this promotion?

10. A bakery sells cakes and cookies. For every cake sold, the bakery earns a profit of ₹20, while for every cookie, it incurs a loss of ₹2 due to a clearance sale. During a special event, the bakery earns a total profit of ₹100 from the sale of 6 cakes and some cookies. How many cookies did it sell during this event?

WORKSHEET - 2

I. MULTIPLE CHOICE QUESTIONS WITH SINGLE CORRECT ANSWER

1. Find the additive inverse of –42. a. –42 b. 42

2. The sum of two integers is 21. If one of them is –6, then the other is_______.

a. –27 b. –15

3. ( ) 316314 −×−×= _____. a. –310 b. 310

4. ( ) 100−÷ =_____.

a. –10 b. 10

5. The value of ( ) ( ) ( ) ( ) ( ) ( ) 0

–62

Not defined

−×−×−×…+−×−×−×… times is _____. a. 105 b. –1

6. Which one of the following pairs of integers does not give the sum equal to –8? a. -6, -2 b. -10, 2 c. 4, -12

7. The value of ( ) ( ) ( ) ( ) ( ) 11121215 −××−××−×−×−× is ______. a. 20 b. –20 c. 14

-4, 4

–14

8. The value of ( ) ( ) ( ) ( ) ( ) ( ) 4141041(4) −+−−−−−+−+−−− is _____. a. 8 b. –8 c. 28 d. –28

9. If ( ) 173 x ÷−=− , then find the value of x.

a. –51 b. –20

10. Find the product of all integers between –3 and 3. a. 4 b. 0

–4

11. The product of three integers is -528. If two of the integers are -11 and -6, then the third integer will be ______.

a. 6

b. 8

c. –8 d. –6

12. The value of ( ) ( ) 200255025 −÷÷÷−   is _____.

a. –4

b. –8

c. 8

d. 4

13. The property represented by the statement (-13) × 0 = 0 × (-13) is _____.

a. Identity property

c. Commutative property

b. Associative property

d. Distributive property

14. The quotient obtained by dividing additive identity by multiplicative identity is ____________.

a. Always 0

b. Always 1

c. Any non-integer d. Always (-1)

15. In a math quiz with 15 questions, each correct answer earns 4 marks, each incorrect answer results in a deduction of 2 marks, and unanswered questions receive 0 marks. Find the score of Rahul, who answered seven questions correctly and four questions incorrectly.

a. 28

b. 20

c. 36 d. 30

16. In a science exam with 20 questions, each correct answer earns 3 marks, each incorrect answer results in a deduction of 1 mark, and unanswered questions receive 0 marks. Find the score of Priya, who answered ten questions correctly and eight questions incorrectly.

a. 22

b. 20

c. 24 d. 18

17. In a debate competition, Team A scored 15, –5, and 10 in three rounds, while Team B scored –10, 20, and –5. Which team scored more overall, and by how much?

a. 5, Team A

b. 15, Team A c. 15, Team B

d. 5, Team B

18. A cup of coffee is served at a temperature of 90°C. Every 5 minutes, the temperature decreases by 3°C. After how much time will the temperature of the coffee be 60°C?

a. 50 mins

b. 55 mins

c. 60 mins

d. 65 mins

19. Divide the sum of (-40), (-50), and (-30) by the product of (–2) and (–3).

a. –5

b. –10

c. –15

d. –20

20. The product of the smallest positive integer and the greatest negative integer is _________.

a. –1

II. FILL IN THE BLANKS

b. 0

1. The difference between 11 and –11 is _____.

c. 1

2. The value of ( ) ( ) ( ) 11124 −×−×−×… times is _____.

3. The sum of (-9), 10 and (-11) is _____.

d. 2

4. The integer between -10 and 2, which is divisible 2 and 3, is _____.

5. Twenty integers are multiplied together. If 13 of them are negative and the rest are positive, then the sign of their product is __________.

6. The value of (-90) ÷ (-15) is _____.

7. The sum of additive inverse of (-15), and multiplicative identity of (-5) is _____.

8. The product of two integers is 182. If one of them is -14, then the other number is _____.

9. The value of (-2) × (-3) × (-2) × 5 is _____.

10. On Monday, the temperature at the location was 22°C. It rose by 5°C on Tuesday, dropped by 7°C on Wednesday, and dropped by 4°C again on Thursday. The temperature at this location on Thursday was _____.

III. SUBJECTIVE QUESTIONS

1. How much is -31 is less than 2?

2. If a = -1 and b = -4, show that (a - b) ≠ (b - a).

3. Combine the subtraction of -420 from -802 with the subtraction of 100 from -117.

4. Write a pair of negative integers whose difference is -12.

5. Evaluate:

i. ( ) ( ) 41(5) −×−×−

ii. ( ) ( ) 3355 −××−×

6. Subtract the product of ( ) ( ) 315,and the product of 5(12). −× −×−

7. In a cryogenic experiment, a substance must be cooled down from 45°C at the rate of 6°C per hour. What will the temperature of the substance be 7 hours into the experiment?

8. A submarine descends into the ocean at a rate of 8 metres per minute. If the descent starts from sea level, which is 0 metres, how long will it take to reach a depth of –240 metres?

9. A clothing store sells shirts and pants. For every shirt sold, the store earns a profit of `15, while for every pair of pants, it incurs a loss of `6 due to clearance. During a clearance sale, the store earns a total profit of `120 from the sale of 8 shirts and some pants. How many pairs of pants did it sell during this sale?

10. Find the quotient obtained by dividing the product of 22 and –15 by the product of –11 and –6.

FRACTIONS AND DECIMALS 2

2.1 FRACTIONS

The number of the form a b , where 0 b ≠ and a, b are whole numbers, is called a fraction.

For fraction a b , a is called the numerator and, b is called the denominator of the fraction.

2.1.1 Types of fractions

Proper fraction

If the numerator of a fraction is smaller than the denominator, then the fraction is called a proper fraction.

Example: 1610 ,, 91130

Improper fraction

If the numerator of a fraction is greater than the denominator, then the fraction is called an improper fraction.

Example: 1134101 ,, 727100

Like fractions

Fractions with the same denominators are called like fractions.

Example: 52344 ,, 777

Unlike fractions

Fractions with different denominators are called unlike fractions.

Example: 316100 ,, 7159

Unit fraction

A fraction whose numerator is 1 is called a unit fraction.

Example: 111 ,, 7159

Mixed fraction

A fraction having a whole number along with a proper fraction is known as a mixed fraction.

Example: 5,3,8642 71113

Equivalent fractions

A given fraction and fraction obtained by multiplying or dividing its numerator and denominator by the same non-zero integer are called equivalent fractions.

Example: 23 , 46

2.2 ADDITION AND SUBTRACTION OF FRACTIONS

2.2.1 Addition of fractions

Case 1: Addition of fractions with equal denominators (like fractions):

To add two fractions with the same denominator, we simply add their numerators and write the common denominator.

Example: 43437 15151515 + +==

Case 2: Addition of fractions with unequal denominators (unlike fractions):

To add two unlike fractions:

• We find the LCM of their denominators.

• Convert the given fractions into like fractions.

• Then add the like fractions.

Example: 511 824 +

Solution:

The given fractions are unlike fractions, so we first find the LCM of their denominators.

2 8, 24

2 4, 12

2 2, 6

3 1, 3 1, 1

LCM of 8 and 24 = 222324 ×××=

Now, we convert the fractions into like fractions (changing the denominator of fractions to 24).

531511 and 832424 × = ×

1511151126 24242424 + +==

2.2.2 Subtraction of fractions

For subtraction, we apply the same rules as in addition for like and unlike fractions.

Case 1: Subtraction of fractions with equal denominators (like fractions):

Example: 1411 2525

Solution:

Here, the given fractions are like fractions. So, we subtract their numerators and keep the denominator the same.

141114113 25252525

Case 2: Subtraction of fractions with unequal denominators (unlike fractions):

Example: 115 1527

Solution:

As the given fractions are unlike fractions, we find the LCM of their denominators.

3 15, 27

3 5, 9

3 5, 3

5 5, 1 1, 1

LCM of 15 and 273335135 =×××=

Next, we convert the fractions into like fractions (fractions with the same denominator).

2.3 MULTIPLICATION OF FRACTIONS

2.3.1 Multiplication of a fraction by a whole number

To multiply a fraction by a whole number, multiply the numerator of the fraction with that whole number keeping the denominator the same.

Example: 151527214 5 and7 444333 ×× ×==×==

2.3.2 Multiplication of a fraction by a fraction

To multiply a fraction by a fraction, multiply their numerators and denominators.

Product of two fractions

Productofnumerators

Productofdenominators =

Example: 14144 272714 × ×== ×

Type 1: To multiply a mixed fraction with a fraction, first convert the mixed fraction into an improper fraction.

Example: 21210 3 5353 ×=× (22) (13) × = × 4 3 = 1 1 3 =

Type 2: If the numerator and denominator have some common factor, first convert it into the simplest form and then multiply.

Example: 416122 1224339 ×=×=

2.4 DIVISION OF FRACTIONS

2.4.1 Division of a whole number by a fraction

To divide a whole number by a fraction, multiply the whole number by the reciprocal of the fraction.

Example: i) 14 3312 41 ÷=×= ii) 33526 255 ÷=×=

To divide a whole number by a mixed fraction, first convert the mixed fraction into an improper fraction and then divide.

2.4.2 Reciprocal of a fraction

Two non-zero fractions whose product with each other gives 1, are called reciprocals of each other.

Example: 32 1 23 ×=

So, 23 is reciprocal of 32 and vice-versa.

Reciprocal can be obtained by interchanging numerator to denominator and denominator to numerator.

Example: For fraction 10 3 , Reciprocal 3 10 =

Reciprocal of a proper fraction is improper and vice-versa.

Example: The reciprocal of 23 32 =

2.4.3 Division of a fraction by a whole number

To divide a fraction by a whole number, multiply the fraction by the reciprocal of the whole number.

Example: 101012 5 3353 ÷=×=

2.4.4 Division of a fraction by another fraction

To divide two fractions, we change the division sign into multiplication and write the reciprocal of the divisor, i.e., multiply the fraction with the reciprocal of the divisor.

Example: 11331124111 48244833236 ÷=×=×=

SOLVED EXAMPLES

Example 1: Solve: 1 153 2 ÷

Example 2: Simplify: 19 5 62 ÷

Example 3: Find:

i. 1 4 of a rupee ii. 3 4 of a day

iii. 7 25 of a kg iv. 2 3 of an hour

Solution:

i. 111 of a rupee = of 100 paise = × 100 paise = 25 paise 444

ii. 333 of a day = of 24 hours = × 24 = 18 hours 444

iii. 777 of kg = of 1000 gm = × 1000 gm = 280 gm 2525 25

iv. 222 of an hour = of 60min = × 60min = 40min 333

Example 4: 25 shirts of equal size were prepared from 5 30m 9 cloth. Find the cloth used for one shirt.

Solution:

Total cloth used 5275 30mm 99 == , Number of shirts = 25

So, cloth required for 1 shirt =

Hence, 2 1m 9 cloth is used for one shirt.

Example 5: Ram cuts 20m long rope into pieces of length 1 3m 3 each. How many pieces of rope did he get?

Solution:

Total length of the rope = 20 m

Length of 1 piece = 110 3mm 33 =

Number of pieces = 103 20206 310 ÷=×=

Hence, Ram got 6 pieces of rope.

Example 6: The product of two fractions is 3 4 . If one of the fractions is 1 7 2 , then find the other.

Solution:

Product of two fractions = 3 4

One of the fractions 115 7 22 ==

Other fraction 315321 4241510 =÷=×=

Hence, the other fraction is 1 10

2.5 DECIMALS

Fractions with denominators in powers of 10, i.e., 10, 100, 1000 etc., are known as decimals. A decimal consists of two parts, that is, whole part and decimal part, which are separated by decimal point (.). The number on the left side of decimal is called whole part and the number formed by the digits at the right side of a decimal is called decimal part. Whole part Decimal part

The number of digits in the decimal part is the number of decimal places.

2.5.1 Like decimals

Decimals having the same number of decimal places are called like decimals.

Example: 11.5, 29.4, 5.2 are like decimals

FRACTIONS

2.5.2 Unlike decimals

Decimals having an unequal number of decimals places are called unlike decimals.

Example: 1.02, 59.1, 49.104 are unlike decimals

Note: Adding zeroes to the extreme right digit of the decimal part does not alter the value of the decimal number. Unlike decimals can be changed into like decimals by adding zeroes to the extreme right of the decimal part. So, 2.400 and 2.712 are like decimals.

2.6 ADDITION AND SUBTRACTION OF DECIMALS

To add or subtract two decimals, we proceed as follows :

Step-1: Write the decimal numbers as like decimals.

Step-2: Write the decimals in columns with their decimal points exactly below each other and other places according to their place value.

Step-3: Add or subtract them as whole numbers.

Step-4: Put the decimals in the answer exactly below the other decimal point.

Example: Add 12.5 + 6.23 Solution:

2.7 MULTIPLICATION OF DECIMAL NUMBERS

We multiply the given decimal numbers without decimal as whole numbers. Then, we mark the decimal point in the product by counting as many places from right to left as the sum of the number of decimal places of the given decimal numbers.

Example: Multiply: i. 2.3 × 4.2

3 × 4.52

And write them in ascending order.

Solution:

i. 2.3 × 4.2

Firstly, multiply the numbers by removing decimals, 2342966 ×=

Now, put the decimal.

2.34.29.66∴×=

ii. 3 × 4.52

3 × 452 = 1356

34.5213.56∴×=

iii. 1.01 × 2.44

Firstly, multiply the numbers by removing decimals.

101 × 244 = 24644

Now, put the decimal.

1.012.442.4644∴×=

Since, 2.4644 < 9.66 < 13.56

1.012.442.34.234.52∴×<×<×

2.7.1 Multiplication of decimal numbers by 10, 100, and 1000

When a decimal number is multiplied by 10, 100, or 1000, the digit in the product remains the same as in the decimal number, but the decimal point is shifted to:

i. 1 place to the right if it is multiplied by 10.

ii. 2 places to the right if it is multiplied by 100.

So, we shift the decimal point to the right in as many places as the number of zeroes is there with one.

Example: Find the following product.

i. 609.75 × 1000 ii. 3.009 × 100 iii. 3.756 × 10

Solution:

i. 609.750 × 1000 = 609750 (Decimal is shifted to three places to the right)

ii. 3.009 × 100 = 300.9 (Decimal is shifted to two places to the right)

iii. 3.756 × 10 = 37.56 (Decimal is shifted to one place to the right)

2.8 DIVISION OF DECIMAL NUMBERS

2.8.1 Division of decimals by 10, 100, 1000 or any multiple of 10

Rule

To divide a decimal by 10,100, 1000, the digits of the number and quotient are the same, but the decimal point of the quotient shifts to the left by as many places as there are zeroes with one.

Example: Divide:

i. 285.6 by 10

Solution:

ii. 2857.9 by 100 iii. 3125.62 by 1000

i. 285.6 ÷ 10 = 28.56 (Decimal is shifted to one place to the left)

ii. 2857.9 ÷ 100 = 28.579 (Decimal is shifted to two places to the left)

iii. 3125.62 ÷ 1000 = 3.12562 (Decimal is shifted to three places to the left)

2.8.2 Division of a decimal number by a whole number

Rule

i. Divide the decimal number by treating it as a whole number by the given whole number.

ii. Put the decimal in the quotient so that it has the same number of decimal places as in the given decimal number.

Example: Divide:

i. 13.6 by 4

Solution:

ii. 14.49 by 7

i. Dividing 136 by 4, we get 34 as the quotient.

So, 13.6 3.4 4 =

ii. 14.49 ÷ 7

Dividing 1449 by 7, we get 207 as the quotient.

So, 14.49 2.07 7 =

2.8.3 Division of a decimal number by another decimal number

Rule

Step 1: We first convert the divisor into a whole number by multiplying the dividend and divisor by 10, 100 or 1000 (as per the number of decimals in the divisor).

Step 2: Now, divide the new dividend by the whole number.

Example: Divide 3.28 by 0.4.

Solution:

Since the divisor contains 1 decimal place, we will multiply the denominator and numerator by 10.

3.283.281032.8 8.2

0.40.4104 × === ×

Thus, 3.28 8.2 0.4 =

SOLVED EXAMPLES

Example 1: If 25 bags of wheat weigh 412.5 kg. Find the weight of 1 bag.

Solution:

Weight of 25 bags of wheat = 412.5 kg

Weight of 1 bag of wheat 412.5 kg16.5kg 25 ==

Example 2: If the cost of 8 pens is ₹ 234.40. Find the cost of one pen.

Solution:

Cost of 8 pens = ₹234.40

Cost of 1 pen = ₹ 23440 8 . = ₹29.30

Example 3: Each side of a regular polygon is 3.5 cm. If its perimeter is 17.5 cm. Find the number of sides of polygon.

Solution:

Number of sides of polygon = 17.5 5 3.5 =

Example 4: The steel needed for the construction of a bridge is 640 tonnes. If the contractor has already purchased 0.65 part of the steel, how many more tonnes of steel must be purchased for the completion of the bridge?

Solution:

Total quantity of steel required = 640 tonnes

This quantity is the total one and whole part required for the construction of the bridge. Out of this, the part already purchased = 0.65

Balance that still needs to be purchased = 1 – 0.65 = 0.35 part

Hence, quantity of steel to be purchased = 0.35 × 640 = 224 tonnes

Example 5: The circumference of the tyre of Radha's bicycle is 2.4 m. She took part in a bicycle race of 1 km. When she had covered 700 m, the tyre burst. If she had to finish the race, how many more times must the wheel go around?

Solution:

The length of the race = 1 km = 1000 m

The distance covered = 700 m

Balance distance = 1000 m – 700 m = 300 m

The circumference of the wheel = 2.4 m

Number of times the wheel must go round = 300 m ÷ 2.4 m = 3000 125 24 =

Thus, the wheel must make 125 more revolutions to enable her to complete the race.

QUICK REVIEW

• The number of the form a b , where b 0 ≠ and a, b are whole numbers is called a fraction. For fraction a b , a is called the numerator and b is called the denominator of the fraction.

• A fraction whose numerator is less than the denominator is called a proper fraction, otherwise it is called an improper fraction.

• Numbers of the type 7953 2 541 18 , , etc., are called mixed fractions

• An improper fraction can be converted into a mixed fraction and vice versa.

• Fractions with the same denominators are called like fractions, and if the denominators are different, then they are called unlike fractions.

• Fractions can be compared by converting them into like fractions and then arranging them in ascending or descending order.

• Sum or difference of like fractions = Sumordifferenceofnumerators Commondenominator

• For adding or subtracting unlike fractions, change them into equivalent like fractions and then add or subtract.

• If and are two fractions, then ac bd ×= acac bdbd .

• If a b and c d are two fractions, then ÷=×= acadad bdbcbc .

• Fractions with denominators in powers of 10, i.e., 10, 100, 1000, etc., are known as decimals.

• While multiplying two decimals, we multiply the numbers without decimals and then put the decimal point in such a way that the number of decimal places in the product is equal to the sum of decimal places of two given numbers.

• To multiply a decimal by 10, 100, 1000 etc., shift the decimal point of the decimal to the right by as many places as the number of zeroes.

• Many daily life problems can be solved by converting different units of measurements such as money, length, weight, etc. in the decimal form and then adding (or subtracting) them.

WORKSHEET - 1

I. Fractions, Addition, Subtraction, Multiplication, and Division of Fractions

1. Match the figures with the given expressions:

2. Match the figures with the given equations:

3. A piece of wire 7 8 metre long broke into two pieces. One piece was 1 4 metre long. How long is the other piece?

4. Manu gave 1 8 part of her money to Tanu. What fraction of money is left with her?

5. Find: a. 1 2 of i. 2 ii. 46 b. 2 3 of i. 18 ii. 27

3 4 of i. 16 ii. 36

1 2 of i. 2 3 4

6. Multiply and express as a mixed fraction:

1 35 5 ×

1 46 3 ×

4 5 of i. 20 ii. 35

3 56 4 ×

7. Jay and Akhil went for a picnic. Their mother gave them a water bottle that contained 5 litres of water. Akhil consumed 2 5 of the water. Jay consumed the remaining water. i. How much water did Akhil drink?

ii. What fraction of the total quantity of water did Jay drink?

8. Which is greater? i. 2335 of or of 7458 ii. 1623 of or of 2737

9. Salima plants 4 saplings in a row in her garden. The distance between two adjacent saplings is 3 m 4 . Find the distance between the first and the last sapling.

10. Neelima reads a book for 3 1 4 hours every day. She reads the entire book in 6 days. How many hours in all were required by her to read the book?

11. A car runs 16 km using 1 litre of petrol. How much distance will it cover using 3 2 4 litres of petrol?

12. Find: i. 3 12 4 ÷ ii. 5 14 6 ÷ iii. 7 8 3 ÷ iv. 1 32 3 ÷ v. 4 53 7 ÷

13. Find the reciprocal of each of the following fractions. Classify the reciprocals as proper fractions, improper fractions, and whole numbers.

i. 3 7 ii. 5 8 iii. 9 7 iv. 6 5

v. 12 7 vi. 1 8 vii. 1 11

14. The length of a rectangular plot of area 2 1 65m 3 is 1 12m 4 . Find the width of the plot.

15. In mid-day meal scheme 1 10 litre of milk is given to each student at a primary school.

If 30 litres of milk are distributed every day in the school, how many students are there in the school?

II. DECIMALS, MULTIPLICATION, AND DIVISION OF DECIMALS

1. Find:

i. 026 × . ii. 846 × . iii. 2.71 × 5 iv. 20.1 × 4

v. 0.05 × 7 vi. 211.02 × 4 vii. 2 × 0.86

2. Find the area of the rectangle whose length is 5.7 cm and breadth is 3 cm.

3. Find:

i. 1.3 × 10

36.8 × 10

v. 31.1 × 100 vi. 156.1 × 100

153.7 × 10

168.07 × 10

3.62 × 100 viii. 43.07 × 100 ix. 0.5 × 10 x. 0.08 × 10

0.9 × 100

0.03 × 1000

4. A two-wheeler covers a distance of 55.3 km in one litre of petrol. How much distance will it cover in 10 litres of petrol?

5. Find:

i. 2.5 × 0.3 ii. 0.1 × 51.7 iii. 0.2 × 316.8 iv. 1.3 × 3.1 v. 0.5 × 0.05 vi. 11.2 × 0.15

1.07 × 0.02

10.05 × 1.05

ix. 101.01 × 0.01 x. 100.01 × 1.1

6. Find:

i. 4.8 ÷ 10 ii. 52.5 ÷ 10 iii. 0.7 ÷ 10 iv. 33.1 ÷ 10

v. 272.23 ÷ 10 vi. 0.56 ÷ 10 vii. 3.97 ÷ 10

7. Find:

i. 2.7 ÷ 100 ii. 0.3 ÷ 100 iii. 0.78 ÷ 100 iv. 432.6 ÷ 100

v. 23.6 ÷ 100 vi. 98.53 ÷ 100

8. Find:

i. 7.9 ÷ 1000 ii. 38.53 ÷ 1000

9. Find: i. 7 ÷ 3.5

128.9 ÷ 1000 iv. 0.5 ÷ 1000

10. A vehicle covers a distance of 43.2 km in 2.4 litres of petrol. How much distance will it cover in one litre of petrol?

WORKSHEET - 2

I. MULTIPLE CHOICE QUESTIONS WITH SINGLE CORRECT ANSWER

1. The expression 141 15153 ++ 

is equivalent to:

2. 0.4 04004 ..×× is equal to a. 6.4

0.64

3. The value of 63728897643766 ..×−×+− is:

0.0064

4. A farmer has 192 animals. 7 16 of animals are cattle. 2 3 of cattle are dairy cows. How many dairy cows does he have?

128 b. 84

5. The quotient when 0.00639 is divided by 0.213 is

112

6. The product of two numbers is 28 27 . If one of the numbers is 5 9 , find the other.

7. The cost of 28 toys of the same kind is ₹ 3462.20, then the cost of each toy is:

₹ 123

₹ 123.65

8. The value of 005005005 ... ×× is

0.000125

₹ 124.65

₹ 122.65

9. Reciprocal of 3 1 4 is

4 1 3

10. By what number should 1 1 2 be divided to get 2 3 ?

2 2 3

11. 0.23 × 0.3 = ? a. 0.69

12. 1.1× 0.1 × 0.01= ? a. 0.011

2 1 3

6.9

0.0011

4 9

0.069

0.11

1 2 4

1.01

13. The perimeter of a square is 1 43cm 6 . Find the length of each side of the square.

2 172cm 3

14. Reciprocal of 57 43 × is

19 10cm 24

1 64cm 2

9 10cm 24

15. In the expression 7.2 × ____ = 0.432, the missing number is a. 0.06

16. If 537 × 2 = 1074, then 537 × 0.02 = ?

0.1074

1.074

10.74

107.4

17. From a wire 1 10 2 metres long, Samir cut off 5 pieces of 1 1 4 metre each. How much length of the wire was left?

18. By selling tickets for ₹52 1 2 each, the money collected was ₹5512 1 2 . Find the number of tickets sold. a. 105 b. 289 c. 106

546

19. The perimeter of a square is 1 43 6 cm. Find the length of each side of the square.

a. 9 11 24 b. 19 11 24 c. 19 10 24

9 10 24

20. The product of a fraction and the sum of 1 3 5 and 1 3 10 is 9 10 . Find the fraction.

a. 7 5 10 b. 9 70 c. 1 7 d. 7

21. Find the reciprocal of the sum of 7 1 9 and 2 1 7

a. 4 3 63 b. 63 256 c. 179 63 d. 63 193

22. 543 hundredths less than 63 tenths is:

a. 48 b. 0.87 c. 60.6

11.73

23. 4 15 of 5 7 of a number is greater than 4 9 of 2 5 of the same number by 8 . What is half of the number?

a. 630 b. 315 c.210 d. 105

24. If 1 02689 3718 . . = , then the value of 1 00003718 . is: a. 2689 b. 2.689 c. 26890 d. 0.2689

25. Find the value of 1 602507503125 4 ...×+− . a. 5.9375 b. 4.2968

II. SUBJECTIVE QUESTIONS

2.1250

1. Find reciprocal of: i. 5 ii. 1 3 iii. 7 9 iv. 1 4 5

2. Divide the sum of 1 3 4 and 3 by 3 2 5

3. Find the multiplication of 4.2, 3.8, and 7.6.

4. The cost of 7.5 kg of rice is ₹262.5. Find the cost of 1 kg of rice.

5. Product of two decimals is 13.25. If one of them is 5.75, find the other decimal.

6. Reenu spends 4 5 of her income on household expenses. Her monthly income is ₹ 15000. How much does she save every month?

7. Simplify: 2712233 17952 ××

8. A tin contains 18 kg ghee. After consuming 2 3 of it, how much ghee is left in the tin?

9. Divide 36 by 2 6 3 and subtract the quotient from 3 7 5 .

10. A car covers a distance of 89.1 km in 2.2 hours. What is the distance covered by it in 1 hour?

III. CASE STUDY

Case: Anup got bored with regular worksheets on fractions and decimals. So, his father planned to play a Snakes and Ladders game and make him practice fractions and decimals worksheets in an interesting way. He said whenever the player encounters a snake they have to solve a question from the worksheet. During the game, Anup encountered snakes 5 times and he solved the following questions.

The value of 0.2 × 20 × 0.02 × 0.2 is:

1. Which of the following fractions is the smallest?

5. Which of the following is correct?

a. 0.45 is greater than 4 5 .

b. 3 4 and 9 4 lies on either side of 3 on the number line.

c. 7 66 and 7 3 9 are reciprocals of each other.

d. None of these

DATA HANDLING 3

3.1 INTRODUCTION

Statistics is a branch of Mathematics which deals with the collection, organisation, presentation, analysis, and interpretation of a collection of numerical figures. In other words, it is a method of expressing one's knowledge through a quantitative understanding of figures. Each numerical figure is called observation, and collection of all observations is called data.

3.2 COLLECTION OF DATA

Let us consider the number of students in classes I to XII in a school: 52, 48, 39, 41, 58, 60, 45, 56, 53, 49, 32, 36.

Here, the number of students does not have a specific arrangement. Such data is called raw data. Each entry in raw data is called an observation.

From the above raw data, it is difficult to answer the questions:

i. Number of girls in a class

ii. Number of boys in a class

To get a better understanding of the data, we arrange the data in ascending or descending order, called an array.

Now, let’s discuss the difference between primary data and secondary data.

Suppose Viraj wants to find who has the maximum height among 10 of his friends in his class, so he prepares a list as follows: 142 cm, 139 cm, 159 cm, 145 cm, 152 cm, 132 cm, 158 cm, 161 cm, 142 cm, 148 cm.

Such data, which is collected by a person for themselves, is called primary data.

If Viraj's friend Anjali needs the same data, she will take it from Viraj and use it for her own purposes. The data which is collected by someone else and used by the investigator for their own requirements is called secondary data.

3.3 ORGANISATION OF DATA

When we collect data, we must record and organise it.

For example, Viraj wants to know the answer to the following questions:

i. Who has the maximum height?

ii. How many of his friends are between 130 cm and 150 cm in height?

To achieve his aim, he organises the data in a tabular form.

From the table, it is clear that Rehan has the maximum height.

To find out how many of his friends have a height between 130 cm and 150 cm, we count the number of friends whose heights fall within this range. By looking at the table, we can see that there are 6 friends of between 130 cm and 150 cm in height.

Now, let us consider another example.

The marks obtained by 30 students of class VII in a class test, out of 50 marks, are as follows:

We can arrange the data in ascending order as follows:

In this example, we notice that a particular observation occurs many times. The number of times a particular observation occurs is called its frequency.

The given information can be arranged in a tabular form, and we can also represent this data with the help of tally marks (|). The number of tally marks indicates the frequency of the observation in the data. If the frequency of an observation is 4 or less than 4, we draw 4 or less than 4 tally marks. However, if the frequency of an observation is 5 or more than 5, we cross the 4 tally marks to indicate 5 and we start again.

Example: The following data shows the number of children in 20 families:

Make a frequency table for the above data.

3.4 RANGE

The range of data is the difference between the largest and the smallest values.

Example: A race was completed by seven participants. Their respective timings to complete the race (in seconds) are given.

13.2, 14.5, 12.9, 13.9, 15.6, 14.1, 12.3

Find the range of their timings.

Solution:

Arranging the data in ascending order:

12.3, 12.9, 13.2, 13.9, 14.1, 14.5, 15.6

Largest value = 15.6; Smallest value = 12.3

Therefore, the range of the given data = 15.6 – 12.3 = 3.3 seconds

3.5 MEASURES OF CENTRAL TENDENCY

There are values that describe the data by identifying the central position within that set of data. These are referred to as measures of central tendency or measures of central location. Three commonly used measures of central tendency are mean, median, and mode.

3.5.1 Arithmetic Mean

Arithmetic mean or mean of a given data is defined as the sum of all the values of the data divided by the total number of values.

Sumofallobservations

Mean= Totalnumberofobservations

Note: Mean is also known as average.

Example: The heights of 5 persons are 144 cm, 152 cm, 151 cm, 158 cm, and 155 cm, respectively. Find the mean height.

Solution:

Sum of the given observations = 144 + 152 + 151 + 158 + 155 = 760 cm

Number of observations = 5

Sumoftheobservations760

Meanheight= ==152cm Total numberofobservations5 ∴

Hence, the mean height is 152 cm.

3.5.2

Median

The median (means middle value) of a group of numbers is the number in the middle, when the numbers are arranged in ascending or descending order.

When the number of collections is odd, then the middle number is the median and when the number of collections is even, then the median is the average of the two middle numbers.

Example: Find the median of the given data: 11, 16, 12, 15, 20, 17, 19

Solution:

Arranging the numbers in ascending order, we have 11, 12, 15, 16, 17, 19, 20.

Now, the number of observations is 7, which is an odd number. So, the median will be the middle number, i.e., 16 is the median.

Example: The marks secured in Mathematics test (out of 25) by 10 students are as follows:

18, 25, 23, 20, 9, 15, 10, 5, 16, 24. Calculate the median.

Solution:

Arranging the observations in ascending order, we have: 5, 9, 10, 15, 16, 18, 20, 23, 24, 25

Here, the number of observations is 10, which is even. So, the two middle numbers are 16 and 18.

Median + ∴=== 161834 17 22

Hence, the median is 17.

3.5.3 Mode

The mode of a group of numbers is the number that appears most often, i.e., the number whose frequency is maximum.

Example: Find the mode if the following numbers of goals were scored by a team in a series of 10 matches.

2, 3, 0, 1, 3, 4, 3, 4, 5, 3

Solution:

Let us prepare a frequency table.

Therefore, the mode is 3, as it occurs most frequently (4 times).

3.6 BAR GRAPH

A bar graph is a pictorial representation of the numerical data using bars (rectangles) of uniform width, erected vertically (or horizontally) with equal spacing between them. The lengths or heights of the bars correspond to the frequency or magnitude of the data being represented.

3.6.1 Construction of a bar graph

The following steps are taken into consideration while constructing a bar graph.

Step 1: Take a graph paper and draw two lines perpendicular to each other that represent horizontal and vertical axes.

Step 2: Along the horizontal axis, take the values of the variables, and along the vertical axis, take the frequencies and label them accordingly.

Step 3: Along the horizontal axis, choose a uniform width for the bars and maintain a uniform gap between the bars, according to the space available.

Step 4: Along the vertical axis, choose a suitable scale to determine the heights of the bars. The scale is chosen according to the space available.

Step 5: Calculate the heights of the bars according to the chosen scale and frequency. Draw the bars.

Example:

Two hundred students from the 6th and 7th classes were asked to name their favourite colour so as to decide what the colour of their school building should be. The results are shown in the following table. Represent the given data on a bar graph.

Answer the following questions with the help of the bar graph:

i. Which is the most preferred colour, and which is the least preferred?

ii. How many colours are there in all? What are they?

Solution:

Choose a suitable scale as follows:

Start the scale at 0.

The greatest value in the data is 55, so end the scale at a value greater than 55, such as 60.

Use equal divisions along the axes, such as increments of 10. You know that all the bars would lie between 0 and 60.

We choose the scale such that the length between 0 and 60 is neither too long nor too small.

Here, we take 1 unit for 10 students.

We then draw and label the graph as shown.

From the bar graph, we conclude that

i. Blue is the most preferred colour (Because the bar representing Blue is the tallest).

ii. Green is the least preferred colour. (Because the bar representing Green is the shortest).

iii. There are five colours. They are Red, Green, Blue, Yellow and Orange. (These are observed on the horizontal line)

3.7 DOUBLE BAR GRAPH

A double bar graph is a graphical representation that allows comparison between two sets of data by using two sets of bars grouped together. It provides information about the relationship between two variables or categories. Let us understand this with an example.

Mr. Khanna asked his students about their interest in different subjects. He found that 20 boys and 15 girls like Mathematics while 10 boys and 12 girls are interested in Science. Also, 14 boys and 10 girls like Social Studies. He asked the students to provide all this information to him properly recorded.

Shikha organised and tabulated the data as follows:

Next, three students, Anuj, Prashant, and Tiya, each made the graphs respectively:

• Anuj made a graph based on the total number of students interested in each subject.

• Prashant made two separate graphs showing the number of boys and girls interested in each subject.

• Tiya made three graphs, each showing the number of students interested in Mathematics, Science, and Social Studies.

Mr. Khanna then explained that all this information could be combined into one graph where the data about boys, girls and all three subjects can be represented, which is called a double bar graph. It tells us about the number of boys and girls interested in each subject and compares the interests of boys and girls in each subject.

Example: The following bar graph shows the number of students who played three different games.

How many more boys like Cricket and Hockey together than girls?

Solution:

Number of boys who like Cricket = 20

Number of boys who like Hockey = 14

∴ Total number of boys who like Cricket and Hockey together = 20 + 14 = 34

Number of girls who like Cricket = 14

Number of girls who like Hockey = 10

∴ Total number of girls who like Cricket and Hockey together = 14 + 10 = 24

∴ Required difference = Total number of boys who like Cricket and Hockey together - Total number of girls who like Cricket and Hockey together = 34 – 24 = 10

Therefore, 10 more boys like Cricket and Hockey together than girls.

SOLVED EXAMPLES

Example 1: Given below is the data showing the number of children in 20 families of a locality: 3, 1, 3, 2, 2, 2, 0, 3, 4, 2, 1, 3, 2, 4, 1, 2, 2, 3, 1, 3.

Arrange the data in ascending order and then prepare a frequency table.

Solution:

Arranging the data in ascending order, we get 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4.

Now, we may prepare the frequency table of the given data, as shown below:

Families

Tally Marks

Example 2: Find the range of 9, 8, 12, 23 and 15.

Solution:

Here, 8 is the smallest number, and 23 is the largest number. Therefore, the range = 23 – 8 = 15.

Example 3: Find the mean of 13, 16.4, 18.8, 9.2, 15.6 and 9.8.

Solution:

Here, Sum of the given observations = 13 + 16.4 + 18.8 + 9.2 + 15.6 + 9.8 = 82.8

Number of observations = 6

Sumofobservations Mean = == Total numberofobservations . . ∴ 828 138 6

Hence, the mean of the given numbers is 13.8.

Example 4: The runs scored by 11 members of a cricket team are 25, 39, 53, 18, 65, 72, 0, 46, 31, 08, 34. Find the median score.

Solution:

Arranging the number of runs in ascending order, we have 0, 08, 18, 25, 31, 34, 39, 46, 53, 65, 72. Here, n = 11, which is odd.

∴ Median score = 6th observation =34.

Hence, the median score is 34.

Example 5: The data given below shows the number of motorcycles of the same brand sold by two dealers in the first six months of a year.

Find the total number of motorcycles sold by dealer II.

Solution:

Number of motorcycles sold by dealer II from Jan to June is as follows:

Jan = 9

Feb = 16

March = 10

April = 3

May = 13

June = 4

Hence,

QUICK REVIEW

• The collection of a particular type of information in numerical figures is called data

• The difference between the maximum and minimum observations of the data is called the range of the data.

• The number of times a particular observation occurs is called its frequency.

• A measure of central tendency represents the centre point or value of a dataset.

• Mean, a most common measure of central tendency, is the sum of all measurements divided by the number of observations in the data, i.e.,

Sumofallobservations

Mean= Total numberofobservations

• If the set of numbers has an odd number of values (i.e., observations), the median is the middle number when all the numbers are arranged in ascending or descending order. If the set of numbers has an even number of values (i.e., observations), the median is the mean of the two middle numbers when all the numbers are arranged in ascending or descending order.

• The value of the observation which occurs most frequently in the data is called the mode

WORKSHEET - 1

I. INTRODUCTION TO DATA HANDLING

1. Given below are the heights (in cm) of 16 girls in a class: 154, 150, 152, 154, 154, 150, 148, 152, 152, 152, 154, 150, 152, 154, 152, 152.

Arrange the data in ascending order and prepare the frequency table.

2. Organise the following marks in a class assessment in a tabular form.

4, 6, 7, 5, 3, 5, 4, 5, 2, 6, 2, 5, 1, 9, 6, 5, 8, 4, 6, 7

i. Which number is the highest?

ii. Which number is the lowest?

iii. What is the range of the data?

iv. Find the arithmetic mean.

3. The heights of 10 students of Class VII are given below (in centimetres): 141, 151, 148, 162, 136, 143, 161, 150, 135, 145

Find the range of the heights.

4. The marks (out of 100) obtained by a group of students in Mathematics test are: 88, 74, 90, 88, 40, 46, 58, 96, 82 and 78.

Find the range of the marks obtained.

II. MEASURES OF CENTRAL TENDENCY

1. The heights of 7 players in a group are 175 cm, 158 cm, 180 cm, 164 cm, 182 cm, 160 cm and 171 cm. Find their mean height.

2. The weights of 10 students (in kg ) are 40, 52, 34, 47, 31, 35, 48, 41, 44, 38. Find the median weight.

3. Given below is the number of pairs of shoes of different sizes sold in a day by the owner of a shop: Size of shoes Number of pairs sold

Find the mode of the given data.

4. The marks obtained by 11 students of a class in a test are given below: 23, 2, 15, 38, 21, 19, 23, 23, 26, 34, 23. Find the mode of the given data.

5. A batsman scored the following runs in six different innings: 37, 40, 50, 49, 60, 58

Calculate the mean of the runs he scored.

6. Find the median of the given data: 10, 16, 13, 15, 22, 18, 19

7. The marks secured in English test (out of 30) by 10 students are as follows: 22, 25, 23, 20, 8, 16, 13, 5, 18, 25

Calculate the median.

8. The marks (out of 100) obtained by a group of students in Mathematics test are: 88, 74, 90, 88, 40, 46, 58, 96, 82 and 78.

Find the median of the marks obtained.

III. BAR GRAPH, DOUBLE BAR GRAPH

1. Read the graph and answer the following questions:

a. What information is conveyed by the bar graph?

b. What was the production of cement in the year 2006-07?

c. During which period was the production minimum?

2. Read the graph and answer the following questions:

Countries / Continents / Subcontinents

a. Give a suitable title to the bar graph.

b. In which part the expenditure on education is maximum in 1990?

c. In which part the expenditure gone up from 1990 to 2000?

3. The production of oil (in lakh tonnes) in some of the refineries in India during 2008 is given below:

Construct a bar graph to represent the above data.

4. The daily maximum and minimum temperature of a city during one week in the month of April is given below. Construct a double bar graph using this data.

WORKSHEET - 2

I. MULTIPLE CHOICE QUESTIONS WITH SINGLE CORRECT ANSWER

1. Which of the following represents statistical data?

a. The names of owners of shops located in a shopping complex.

b. A list giving the names of all states of India.

c. A list of all European countries and their respective capital cities.

d. The volume of rainfall in a certain geographical area recorded every month for 24 consecutive months.

2. Which measure of central tendency best describes the data of the most demanding size of shoes after sale?

a. Mean b. Mode c. Median d. Range

3. Runs scored by a batsman in four matches are 27, 83, 64 and 56. If he scores 75 runs in the fifth match, then his average score will be:

a. Increase by 3.5 b. Decrease by 3.5 c. Increase by 2 d. Decrease by 2.5

4. The following bar graph shows the rainfall at selected locations in certain months.

Which of the following statements is correct?

a. July rainfall exceeds August rainfall by 100 cm in each location.

b. September rainfall exceeds August rainfall by 50 cm in each location.

c. July rainfall is lower than August rainfall in each location.

d. None of these

5. From a series of 50 observations, an observation with the value of 45 is dropped, but the mean remains the same. What was the mean of 50 observations?

a. 50 b. 49

6. The average marks in Maths of 100 students in a class was 72. The mean marks for boys was 75, while their number was 70. The mean marks of girls in the class was:

a. 35

7. The numbers 4 and 9 have frequencies x and (x – 1) respectively. If their arithmetic mean is 6, then x is equal to:

8. If the median of x 5 , x, x 4 , x 2 and x 3 (where x > 0 ) is 8, then the value of x 2 would be: a. 24

9. The mean of 12, 10, x, 20 and 18 is 17. Find x. a. 12

10. The bar graph given below shows the interest of boys and girls in different activities.

Scale: 1 unit = 5 students

In which two activities are 55 girls interested together? a. Art and music b. Music and dance c. Dance and yoga d. Yoga and dramatics

II. FILL IN THE BLANKS

1. In the data given below, the observation p is missing. 2.5, 1.5, p, 1.7, 2.5, 1.3, 3.5

If the mean is 2.5, then the missing observation p will be ________.

2. The mean of 11 numbers is 7. If each number is multiplied by 6, then the mean of the new set of numbers is ________.

3. In a class, there are 20 girls and 30 boys. The mean weight of 20 girls is 35 kg, and the mean weight of 30 boys is 60 kg. The mean weight of the class will be ________.

4. If the mean of 2, 3, x, 7, and 8 is x, then the value of x will be ________.

5. The heights of 7 students (in cm) are given as: 120, 126, 132.4, 121.5, 120.3, 132, 125. The median height of the students will be ________.

III. SUBJECTIVE QUESTIONS

1. Find the mean of the first five prime numbers.

2. Find the mode of the data 7, 8, 9, 9, 10, 7, 11, 10, 7, 6.

3. Find the median of data 9, 12, 11, 10, 8, 9.

4. The table given below shows the expenditures on different components of a person.

Represent the data in a bar graph.

5. The marks (out of 100) obtained by a student in his quarterly examination in various subjects are given below:

Represent the data in a bar graph.

SIMPLE EQUATIONS 4

4.1 INTRODUCTION

1. A symbol which can take various numerical values is called a variable. Variables can be represented as alphabetical letters such as a, b, c, x, y, z, etc.

2. A symbol which has a fixed value is called a constant.

3. The constant alone or the variable alone or their combinations by operation of multiplication or division are called terms.

4. Expressions of the form 7 + 5, 8 - 3, 24 × 6, 22 3 , etc., are called numerical expressions.

5. If two numerical expressions are joined or connected by 'is equal to' (=) or 'is greater than' (>) or 'is less than' (<), etc., then they are called mathematical sentences.

6. A mathematical sentence that can be verified as true or false, but not both, is called a mathematical statement.

4.2 What Is an Equation?

An equation is a condition on a variable. The condition is that two expressions must be equal.

Note: At least one of the two expressions must contain the variable. Or

An equation is a statement which states that the two expressions are equal. coefficient expression Terms: 5x, 7, 2

5x + 7 = 2 expression equation variable constant

Examples: i. If the expressions 3 + 2 x and 7 x are equal, then 327 xx+=− forms an equation.

ii. The expressions 8y – 15 and 2 y are equal 815 2 y y ⇒−= is an equation.

The linear equation which involves one variable is called a linear equation in one variable or simple equation.

Examples: i. 352 xx−=+ ii. 2 10 y +=

Note: ax + b = 0 is the general form of a linear equation in one variable, where a () 0 ≠ and b are real numbers.

LHS and RHS notations: The sign of equality (=) in an equation divides it into two sides, namely, the left-hand side and the right-hand side, written as LHS and RHS, respectively.

Example: ,352 xx−=+

Here, LHS = 35 x

RHS = x + 2

4.3 SOLVING AN EQUATION

4.3.1 Solution of an equation

Solving linear equations involves finding the value of an unknown algebraic quantity (variable) present in the equation.

Examples: i. To solve the equation x + 5 = 7 means finding the value of x.

ii. To solve the equation 3y + 2 = 9 means finding the value of y.

iii. To solve the equation 2 410 3 a a += means finding the value of a and so on.

The replacement value of a variable in an equation that makes LHS = RHS is called the solution or the root of the equation.

Example: 47 x +=

Solution:

Here, LHS = x + 4

RHS = 7

The equation is true only when x = 3.

i.e., 3LHS347 x =⇒=+=

RHS = 7

LHSRHS∴=

The solution/rootof47is3 x ∴ +=

4.3.2 Solving an equation

The method of finding the solution for a given equation is called solving an equation.

An equation remains unchanged if:

1. The same number is added to both sides of the equation.

i.e., 56 x −=

5565 x ⇒−+=+ [Adding 5 on both the sides]

11 x ⇒=

2. The same number is subtracted from both sides of the equation.

i.e., 56 x +=

5565 x ⇒+−=− [Subtracting 5 from both the sides]

1 x ⇒=

3. The same number is multiplied on both sides of the equation.

i.e., 2 3 x = 323 3 x ⇒×=× [Multiplying both the sides by 3]

6 x ⇒=

4. Each side of the equation is divided by the same non-zero number.

i.e., 412 x = 412 44 x ⇒= [Dividing both the sides by 4] 3 x ⇒=

SOLVED EXAMPLES

Example 1: Write the following statements in the form of equations:

i. The sum of three times x and 11 is 32.

ii. If you subtract 5 from 6 times a number, you get 7.

Solution:

i. Three times x is 3x

The Sum of 3x and 11 is 3x + 11.

The sum is given as 32.

Thus, the equation is 3x + 11 = 32.

ii. Let us say the number is z; z multiplied by 6 is 6z.

Subtracting 5 from 6z, one gets 6z – 5.

The result is given as 7.

Thus, the equation is 6z – 5 = 7.

Example 2: Solve the equation x + 3 = 9.

Solution: x + 3 = 9

3393 x ⇒+−=− [Subtracting 3 from both the sides]

6 x ⇒=

Example 3: Solve: 2511 x +=

Solution:

2511 x +=

255115 x ⇒+−=− [Subtracting 5 from both the sides]

26 x ⇒=

26 22 x ⇒= [Dividing each side by 2]

3 x ⇒=

SOLVING A SIMPLE EQUATION USING TRANSPOSITION

METHOD 4.4

1. In transposition (shifting), a positive term becomes negative.

i.e., x + 3 = 7

73 x ⇒=−

4 x ⇒=

2. Also, in transposition (shifting), a negative term becomes positive.

i.e., - 1 = 5 x

51 x ⇒=+

6 x ⇒=

3. In an equation, if a term in multiplication is transposed to the other side, its sign is reversed, i.e., it becomes division.

i.e., 3x = 6

x ⇒= [Transposing 3, which is in multiplication with x] 2 x ⇒=

4. In an equation, if a term is in division, it is transposed to the other side in multiplication.

i.e., 6 3 x = 63 x ⇒=× [Transposing 3, which is in division with x] 18 x ⇒=

Note: Although it is not a rule, the variable (x, y or z, etc.) is preferred to be kept on the left-hand side of the equation.

4.5 SOLVING AN EQUATION WITH VARIABLES ON BOTH SIDES

Transpose the terms containing the variable to one side and the constants (i.e., terms without the variable) to the other side.

Example: Solve: 8359 xx−=+

Solution:

Given, 8359 xx−=+

Transposing + 5x to the left and -3 to the right side, we get: 8593 xx−=+

Example 1: Solve the following equations:

i. 2 16 3 x = ii. 3 58 4 x += iii. 13 5 24 x −=

Solution:

i. 2 16 3 x =

45 x ⇒= × [Transposing 5] 1 4 x ⇒=

Example 2: Solve: ()()() 2532874 xxx −+−=+−

Solution:

Given, ()()() 2532874 xxx −+−=+− 210368728 xxx ⇒−+−=+− 516720 xx ⇒−=− 572016 xx ⇒−=−+ 24 x ⇒−=−

4.6 APPLICATIONS OF SIMPLE EQUATIONS

4.6.1

Solving word problems

1. Read the given problem carefully to know what is given and what we need to find.

2. Represent the quantity required to find by letter x or by letter y (other letters such as a, b, c, ..., etc., can also be taken instead of x and y).

3. According to the condition(s) given in the problem, write the relation (equation) between the knowns and the unknowns.

4. Solve the equation to obtain the value of the unknown.

Example: If 8 is added to twice a certain number, the sum is 36. Find the number.

Solution:

Let the required number be x.

When 8 is added to twice of x, we get 2x + 8.

Since this sum is equal to 36:

2836 x ⇒+=

2368228 and 14 xxx ⇒=−⇒==

⇒ 2368228 and 14 xxx ⇒=−⇒==

∴ The required number is 14.

SOLVED EXAMPLES

Example 1: Three times a certain number, diminished by four, equals 20. Find the number.

Solution:

Let the number be x.

Given that, 3 times the number diminished by 4 equals 20.

3420 x ∴−=

3204 x ⇒=+ [Transposing 4] 24 8 3 x ⇒==

∴ The required number is 8.

Example 2: Find four consecutive integers such that one-third of the smallest exceeds one-sixth of the largest by 2.

Solution:

Let the required integers be x, x + 1, x + 2, and x + 3.

Given that, () 11 of of 32 36 xx−+= () 11 32 36 xx ×−×+=

123 x ⇒=+

15 x ⇒=

∴ The required integers are 15, 15 + 1, 15 + 2, and 15 + 3, i.e., 15, 16, 17, and 18.

Example 3: Raju's father's age is 5 years more than three times Raju's age. Find Raju's age, if his father is 44 years old.

Solution:

Let the age of Raju be y years.

Three times Raju's age is 3y years.

Given that, Raju's father's age is 5 years more than 3y

Thus, Raju’s father's age = (3y + 5) years old.

It is also given that Raju's father is 44 years old.

Therefore, 3y + 5 = 44

To solve it, we first transpose 5 to RHS:

3 = 44 - 5 y ⇒

3 = 39 y ⇒ = 13 y ⇒

Hence, Raju's age is 13 years.

Example 4: Rajesh had some money. He gave one-fourth of it to his friend Rohit and still has ₹144 left with him. Find, how much money he had in the beginning.

Solution:

Let x represent the amount of money Rajesh had initially.

So, he gave ₹ 4 x to his friend Rohit. 144 4 x x ∴−= 4 144 4 xx ⇒=

31444 x ⇒=× 1444 192 3 x × ⇒==

∴ Rajesh initially had ₹192.

Example 5: One number is four times another number. If the larger number is subtracted from 100, the result is 10 less than the number obtained by subtracting the smaller number from 50. What are the two numbers?

Solution:

Let the smaller number be x.

So the larger number = 4x

100 – 4x represents the larger number subtracted from 100.

50 – x represents the smaller number subtracted from 50.

As per the condition, () 10045010 xx −=−− 100440xx⇒−=− 100404xx⇒−=−+ 603 x ⇒= 20 x ⇒= , 20480xx ∴==

Hence, the smaller number is 20, and the larger number is 80.

QUICK REVIEW

• An equation is a condition on a variable such that two expressions in the variable should have equal value.

• The value of the variable for which the equation is satisfied is called the solution of the equation.

• An equation remains the same if the LHS and the RHS are interchanged.

• In the case of the balanced equation, if we:

i. add the same number to both sides, or

ii. subtract the same number from both sides, or

iii. multiply both sides by the same number, or

iv. divide both sides by the same number; the balance remains undisturbed, i.e., the value of the LHS remains equal to the value of the RHS.

• Transposing a term involves moving it from one side of an equation to the other, applying the opposite operation (addition becomes subtraction, subtraction becomes addition, multiplication becomes division, division becomes multiplication) while maintaining equality.

WORKSHEET - 1

I. SOLVING AN EQUATION

1. Solve the following simple equations.

i. 743xx =−

iii. 7330 tt −=−

v. ( ) ( )33521 tt−=+

ii. 412362 pp +=−

iv. 95151 yy+=−

vi. ()()() 15429560 ttt −−−++=

II. SOLVING AN EQUATION USING TRANSPOSITION AND VARIABLE ON BOTH SIDES

1. Find the value of x, if x - 3 = 7.

2. Find the value of m, if . 11 43 m +=

3. Solve for m: 5m = 75

4. Solve for x:

i. 1 38 4 x −=

ii. 2 15 3 x +=

5. Solve: ()()() 2532874 xxx −+−=+−

6. Solve: 213 725 x x + =

7. Solve: 53322 233 xx −=

8. Solve: 111 22 43 yy+=+

9. Solve: 53322 233 xx −=

10. Solve: 33525 25 753 xxx +−− −=−

III. APPLICATIONS OF SIMPLE EQUATIONS

1. One number is 3 less than two times the other. If their sum is increased by 7. The result was 37. Find the number.

2. There are some lotus flowers in a pond and some bees are hovering around. If one bee lands on each flower, one bee will be left; if two bees land on each flower, one flower will be left. Find the number of flowers and bees.

3. A positive number is 5 times another. If 21 is added to both numbers, then one of the new numbers becomes twice the other new number. What are the numbers?

4. The sum of the digits of a two-digit number is 9. When we interchange the digits, it is found that the resulting new number is greater than the original number by 27. What is the twodigit number?

5. One of the two digits of a two-digit number is three times the other digit. If you interchange the digits of this two-digit number and add the resulting number to the original number, you get 88. What is the original number?

6. Shobu’s mother's present age is six times Shobu’s age. Five years from now, Shobu’s age will be one-third of his mother's present age. What are their present ages?

7. A grandfather is ten times older than his granddaughter. He is also 54 years older than her. Find their present ages.

8. A man is 35 years old, and his son is 7 years old now. In how many years will the son be half as old as his father?

9. The present age of A is twice that of B. 30 years from now, the age of A will be 1 1 2 times that of B. Find the present ages of A and B.

10. If 4 more than 8 times a number is 60. Then, find the number.

11. Monica subtracts thrice the number of notebooks she has from 100. She finds the result to be 7. Find the number of notebooks she has.

12. The sum of three consecutive integers is 10 more than twice the smallest of the integers. Find the integers.

13. After buying a toy from the shop, Rohan left with ₹ 15 of his 100 rupees of pocket money. What is the cost of the toy?

14. A number is multiplied by 3, and 7 is taken away from the product to get the answer 17. What is the number?

15. Ravina's weight is 4 kg more than twice her sister's weight. If Ravina's weight is 50 kg, then find her sister's weight.

16. There are two sections of class VII in a school. The number of students in section A is 4 less than thrice the number of students in section B. If section A has 56 students, then find the number of students in section B.

17. Rampal owns a rectangular plot of land. Its breadth is 7 m less than its length. The perimeter of the plot is 1034 m. Find the length and breadth of the plot.

18. Divide 64 into two parts such that one part is 3 times the other.

19. In a cricket match, Rajat scored thrice as many runs as David. Together, they fell short of 8 runs to a triple century. Find their scores.

20. The teacher tells the class that the highest marks obtained by a student in her class are twice the lowest marks plus 7. The highest score is 87. What is the lowest score?

21. In an isosceles triangle, the base angles are equal. The vertex angle is 40° What are the base angles of the triangle?

22. Sachin scored twice as many runs as Rahul. Together, their runs fell two shorts of a double century. How many runs did each one score?

WORKSHEET - 2

I. MULTIPLE CHOICE QUESTIONS WITH SINGLE CORRECT ANSWER

1. The standard form of a simple equation in one variable x is:

a. 0 axb+= b. 2 0 axbxc++=

c. 32 0 axbxcxd+++= d. 432 0 axbxcxdxe ++++=

2. The equation form of the statement, 'a number when added to 10, becomes 20' is:

a. 1020 x −= b. 1020 x += c. 1020 x = d. 20 10 x =

3. Seven times a number is 42. This statement in the form of an equation is:

a. 742 x += b. 742 x = c. 42 7 x = d. 742 x −=

4. The root of the equation, 20 3 7 xx =− is:

a. 10 7 b. 20 21 c. 5 7

5 7

5. The largest number of the three consecutive numbers is x + 1, then the smallest number is:

a. x + 2

x + 1

x – 1

6. If x is an even number, then the consecutive even number is:

a. x + 1 b. x + 2 c. 2x d. x + 3

7. If x is an odd number, the largest odd number preceding x is:

a. x – 1 b. x – 2 c. x – 3 d. x – 4

8. The difference between the two numbers is 21. If the larger number is x, then the smaller number is:

a. 21 + x b. x-21 c. 21 – x d. –x –21

9. If , 1 43 322 xx +=+ then x =?

a. 1 3 b. 1 3 c. -3 d. 3

10. If 0.5x - 2.5 = 77.5 - 0.3x, then x = ___.

a. 80 b. 10 c. 100 d. 100 8

11. If ()()() , 3111 3 522 xxx −+− −=− then x = ___.

a. 2 b. 3 c. 7 d. 12 7

12. If the sum of three numbers is 60, and their ratio is 1 : 2 : 3, then the largest number is:

a. 60 b. 10 c. 20 d. 30

13. The length of a rectangle is 4 cm more than its breadth. If the perimeter of the rectangle is 4.16 m, then its breadth is:

a. 102 m b. 1.02 m c. 1020 cm d. 104 cm

14. If the perimeter of a triangle is 18 m, and the ratio of their sides is 2 : 3 : 4, then the smallest side is:

a. 2 m b. 4 m c. 6 m d. 8 m

15. If the sum of four consecutive integers is 46, then the four integers are:

a. 11, 12, 13, 14 b. 10, 11, 12, 13 c. 6, 7, 8, 9 d. 20, 21, 22, 23

16. If a number is tripled and the result is increased by 5, we get 50, then the number is:

a. 10 b. 25 c. 20 d. 15

17. If a number is such that it is as much greater than 84 as it is less than 108, then the number is:

a. 86 b. 96 c. 106 d. 116

18. If a number is 56 greater than the average of its one-third, quarter, and one-twelfth, then the number is:

a. 62 b. 82 c. 72 d. 92

19. If 0.16(5x – 2) = 0.4x + 7, then x = ___.

a. 16.3 b. 17.3 c. 18.3 d. 20.3

20. If 8175 7 368 xx x +−=− , then x = ___.

a. 4 b. 5 c. 6 d. 7

21. 3 4 part of a number is 5 more than its 2 3 part. This statement in the form of an equation is:

a. 23 5 34 xx−=

b. 23 5 34 xx −=

c. 32 5 43 xx=+ d. 32 5 43 xx −=

22. On adding 9 to the twice of a whole number gives 31. The whole number is:

a. 21 b. 16 c. 17 d. 11

23. The sum of two consecutive odd numbers is 36. The smaller one is:

a. 15 b. 17

24. Thrice of a number when increased by 6 gives 24. The number is:

a. 6 b. 7 c. 8 d. 11

25. If 2m – 6 = 14, then the value of 3m – 6 is;

a. 18 b. 16 c. 12 d. 24

26. Solve for x: 18 - (2x - 12) = 8x

a. 15 b. 26

27. Solve for m: ( ) ( ) 3222025 mmm ++=−−

II. FILL IN THE BLANKS

c. 3

13 5

1. Twice of a number is as much greater than 30 as three times the number less than 60. The number is ______________.

2. One number is greater than the other number by 3. The sum of the two numbers is 23. The two numbers are ___________.

3. If 9 is added to twice a number, we get 67. The number is ____________.

4. The root of the equation (2x - 1) + (x - 1) = x + 2 is ______________.

5. The base of an isosceles triangle is 6 cm, and its perimeter is 16 cm. The length of each of the equal sides is______________.

6. If cx + d = 0, then x =______________.

7. If , 7 1 315 x += then x =______________.

8. The sum of a two-digit number and the number obtained by reversing the digits is a multiple of_______________.

III. SUBJECTIVE QUESTIONS

1. Write each of the following statements as an equation:

i. Five times a number x is equal to 30.

ii. If 7 is taken away from the number y, the result is 4.

iii. Ten more than y is 15.

iv. Thrice a number x decreased by 5 is equal to 27.

v. 20 decreased by a number x is equal to 15.

vi. x decreased by 5 is 12.

2. Solve the following equation and check your answer. 710440 xx+=+

3. Solve the following equation and check your answer: 62222 yy−=+

4. Solve: 5436 854 xxx−−+ −=

5. Solve: ()() 219 52 342 xx−−−=

6. The two equal sides of an isosceles triangle are 4x - 2 and 3x + 4. If the third side is 2x - 1,

then find each side and the perimeter of the triangle.

7. The length of a rectangle is three times its width. If the perimeter of the rectangle is 96 m, then find the length and breadth of the rectangle.

8. There are only 50 paise coins in a purse. If the total value of the money in the purse is ₹125, then find the number of coins in the purse.

9. After 15 years, Vikas will be four times as old as he is now. Find his present age.

10. Rakhi is 26 years younger than her mother. After 9 years, Rakhi's mother will be twice as old as Rakhi. Find the present age of Rakhi in years.

11. Find x, if 51311 xx+=+ .

LINES AND ANGLES 5

5.1 INTRODUCTION

The corners made by the intersection of two lines or line segments are called angles.

We write the angle as ∠ABC in the first figure and ∠XOY, ∠ZOW, ∠YOW and ∠XOZ are angles in the second figure.

5.1.1

Acute angle

An angle whose measure is less than 90° is called an acute angle.

Example: 30°, 60°, 70° etc.

5.1.2

Right angle

An angle whose measure is 90° is called a right angle.

5.1.3 Obtuse angle

An angle whose measure is greater than 90° and less than 180° is called an obtuse angle.

Example: 120°, 135°, 140° etc.

Obtuse angle

5.1.4 Straight angle

An angle whose measure is 180° is called a straight angle.

5.1.5 Reflex angle

An angle whose measure is more than 180° and less than 360° is called a reflex angle.

Example: 220°, 345°, 190° etc. Reflex angle

5.2 RELATED ANGLES

5.2.1 Complementary angles

Two angles are said to be complementary if their sum is 90° .

Example: 30° and 60° are complementary angles.

5.2.2 Supplementary angles

Two angles are said to be supplementary if their sum is 180°

Example: 70° and 110° are supplementary angles.

5.2.3 Adjacent angles

Two angles are said to be adjacent if

i. they have a common vertex

ii. they have a common arm

iii. their non-common arms on either side of the common arm

Here ∠ABD and ∠DBC are adjacent angles.

5.2.4 Linear pair of angles

A linear pair is a pair of adjacent angles whose non-common sides are opposite rays. The angles in a linear pair are supplementary.

Here, ∠AOC and ∠BOC form a linear pair.

5.2.5

Vertically opposite angles

Two angles are called a pair of vertically opposite angles, if their arms form two pairs of opposite rays. Here,

i. ∠AOC and ∠BOD form a pair of vertically opposite angles.

ii. ∠BOC and ∠AOD form a pair of vertically opposite angles.

Note: Vertically opposite angles are equal.

SOLVED EXAMPLES

Example 1: How many degrees are there in:

i. two right angles?

ii. 1 6 right angle?

iii. 1 1 2 right angle?

iv. a straight angle?

v. 2 5 straight angle?

Solution:

i. Two right angles = 2 × 90° = 180°

ii. 1 6 right angle = 1 6 × 90° = 15°

iii. 1 1 2 right angle = 3 2 right angle = 3 2 × 90° = 3 × 45° = 135°

iv. A straight angle = 180°

v. 2 5 straight angle = 2 5 × 180° = 2 × 36° = 72°

Example 2: If x and (x + 30°) are complements of each other, then find the value of x.

Solution:

Let x and x + 30° are complements of each other.

∴ x + x + 30° = 90°

⇒ 2x + 30° = 90°

Subtracting 30° from both sides, we get

2x = 90° - 30°

⇒ 2x = 60°

⇒ x = 30°

Example 3: If z and ( ) o 50 z + are supplements of each other, then find the value of z.

Solution:

Let z and z + 50° are supplements of each other.

⇒ z + z + 50° = 180°

⇒ 2z + 50° = 180°

Subtracting 50° from both sides, we get

2z + 50° - 50° = 180° - 50°

⇒ 2z = 130°

⇒ z = o o 130 65 2 =

∴ z = 65°

Example 4: An angle is double its complement. Find the angle.

Solution:

Let the angle be x.

Its complement = double of x = 2x

∴ x + 2x = 90°

⇒ 3x = 90°

Dividing both sides by 3, we get

x = o o 90 30 3 =

∴ One angle = 30° and other angle = 2 × 30° = 60°

Example 5: The measure of an angle is 30° less than its supplement. Find the measure of the angles.

Solution:

Let one of the angles be x

Measure of its supplement = x - 30°

∴ x + x - 30° = 180° ⇒ 2x - 30° = 180°

Adding 30° to both sides, we get

2x = 180° + 30° ⇒ 2x = 210° ⇒ x = 105°

∴ One angle = 105°

Its supplement = 105° - 30° = 75°

∴ Measure of both angles are 105° and 75° .

Example 6: Find an angle which is two fifths of its supplement.

Solution:

Let the angle be x.

∴ Its supplement = 180° - x

According to the question,

x = 2 5 (180° - x)

⇒ 5x = 360° - 2x

⇒ 7x = 360°

∴ x = 360 7

Example 7: In the given figure, AOB is a straight line. Find

∠ AOC

2x + 25o x + 35o

Solution:

Since ∠ AOB is a straight line.

∴ x + 35° + 2x + 25° = 180°

⇒ 3x + 60° = 180°

⇒ x = 40°

Thus,

⇒ ∠ AOC = 40° + 35° = 75° And,

⇒ ∠ BOC = 2(40°) + 25°

⇒ ∠ BOC = 105°

∠ BOC

5.3 PAIRS OF LINES

Pairs of lines are two lines that are related in a geometric context, often by their positions and angles relative to each other. Examples include parallel lines, perpendicular lines, and intersecting lines.

5.3.1 Intersecting lines

Intersecting lines are lines that meet or cross each other at a common point known as the point of intersection, forming angles at this intersection. They do not have the same direction and can intersect at any angle.

Here, lines l and m intersect each other at point O.

5.3.2 Transversal

A line which intersects two or more given lines at distinct points is called a transversal of the given lines.

In the above figure, AB and CD are given two lines and a transversal LM intersects them at points P and Q, then eight angles are formed.

5.3.3 Transversal of parallel lines

A transversal of parallel lines is a line that intersects two or more parallel lines. It forms corresponding angles, alternate interior angles, alternate exterior angles, and consecutive interior angles with the parallel lines.

Parallel lines = a,b

Transversal = l

5.3.4 Angles made by a transversal

Alternate angles

Two angles are considered to be a pair of alternate angles if,

a. they are on either side of the transversal

b. both are interior angles or exterior angles

c. they are not adjacent angles

Note: If alternate angles are equal then the lines are parallel.

Alternate interior angles

A pair of angles on opposite sides of the transversal but inside the two lines are called alternate interior angles.

By the alternate interior angles theorem, the pairs of alternate interior angles in the above figure are:

∠4 and ∠6

∠3 and ∠5

Alternate exterior angles

The pairs of angles on opposite sides of the transversal but outside the two lines are called alternate exterior angles.

The pairs of alternate exterior angles in the above figure are:

∠1 and ∠7

∠2 and ∠8

Corresponding angles

Two angles are said to be a pair of corresponding angles if

a. they are on the same side of the transversal.

b. one is the interior angle and the other is the exterior angle.

c. they are not adjacent angles.

According to the given figure,

∠1 = ∠2 (corresponding angles)

Note:

If a transversal intersects two parallel lines, then:

• each pair of corresponding angles are equal.

• each pair of alternate interior angles are equal.

• each pair of interior angles on the same side of the transversal is supplementary.

If a transversal intersects two lines, then the lines are parallel if they are:

• making a pair of corresponding angles that are equal.

• making a pair of alternate interior angles that are equal.

• making a pair of interior angles on the same side of the transversal that are supplementary.

SOLVED EXAMPLES

Example 1: Find the measure of x in the given figure, if l || m l

Solution:

As l || m

∴ ∠ 3 = x [Alternate interior angles]

Now, ∠ 4 + ∠ 3 = 180° [Linear pair]

 ∠ 4 = 4x and ∠ 3 = x

∴ 4x + x = 180°

⇒ 5x = 180° ⇒ x = 36°

Example 2: In the given figure, AE || GF || BD, AB || CG || DF and ∠ CHE = 120°. Find ∠ ABC and ∠ CDE

C B 120o

Solution:

Since, AE || BD and CH is a transversal.

∴ ∠ CHE = ∠ HCB = 120° ... (i) [Alternate interior angles]

Now, CH || DF and CD is a transversal.

∴ ∠ HCB = ∠ CDE [Corresponding angles]

⇒ ∠ CDE = 120° ...(ii) [Using (i)]

Also, AB || DF and BD is a transversal.

∴ ∠ ABC + ∠ CDE =180° ... [Co-interior angles]

⇒ ∠ ABC = 180° - 120° ...[Using (ii)] = 60°

Thus, ∠ABC = 60° and ∠CDE = 120°

Example 3: In the given figure, AB || CD and EF is a transversal. If ∠AGE = 110°, then find x, y and z.

Solution:

We have, ∠ AGE = 110°

x = 110° [Vertically opposite angles]

Given that, AB || CD, ⇒ ∠x= ∠y= 110° [Alternate interior angles]

Since, the sum of co-interior angles is 180° .

∴ x + z = 180°

⇒ 110° + z = 180°

⇒ z = 180° - 110° = 70°

∴ x = 110° , y = 110° and z = 70°

Example 4: In the given figure AB || CD, ∠CDM = 150° , BMD = 64° and ∠ABM = x. Find the value of x

Solution:

Through M, draw EMF || CD || AB

Let ∠ EMB = y

Then, ∠ EMD = 64° - y

Now EF || CD and MD is a transversal

∴ The sum of co-interior angles is 180° .

∴ 150°+ 64° - y = 180°

⇒ y = 150° + 64 - 180° = 34°

Again AB || EF and BM is a transversal.

∴ The sum of co-interior angles is 180° .

∴ x + y = 180°

⇒ x + 34° = 180°

⇒ x = 180° - 34° = 146°

Example 5: In the figure given below, determine whether AB || CD

Solution:

∠GHD =∠CHF = 55° [Vertically opposite angles]

Now, ∠ BGH + ∠GHD = 125° + 55° = 180°

Thus, the sum of co-interior angles is 180°

Hence, AB || CD

QUICK REVIEW

• Two angles are said to be complementary, if the sum of their measure is 90°

• Two angles are said to be supplementary, if the sum of their measure is 180° .

• Two angles having a common arm and a common vertex such that the other arms of the angles are on the opposite sides of the common arm, are known as adjacent angles.

• When two lines intersect, then two angles having no common arm are known as vertically opposite angles.

• Two adjacent angles are said to form a linear pair, if these angles have one arm common and the other arms as opposite rays.

• If a ray stands on a line, then the sum of the two adjacent angles so formed is 180° and vice versa. This property is called the linear pair axiom.

• If two lines intersect each other, then the vertically opposite angles are equal.

• If a transversal intersects two parallel lines, then

i. each pair of corresponding angles are the same,

ii. each pair of alternate interior angles are the same,

iii. each pair of interior angles on the same side of the transversal is supplementary.

• If a transversal intersects two lines such that, either

i. any 1 pair of corresponding angles are the same, or

ii. any 1 pair of alternate interior angles are the same, or

iii. any one pair of interior angles on the same side of the transversal is supplementary, then the lines are parallel.

• Lines parallel to a given line are parallel to one another.

WORKSHEET - 1

I. INTRODUCTION AND RELATED ANGLES

1. Find the complement of each of the following angles:

2. Find the supplement of each of the following angles:

3. Find if the following pairs of angles are supplementary, complimentary or none of these.

4. The angle which is the same as its complement is ________________.

5. The angle which is the same as its supplement is ________________.

6. ∠1 and ∠2 are supplementary angles in the following image. If ∠1 is decreased, then give the conditions for ∠2 so that both the angles still remain supplementary.

7. Can two angles be supplementary if both of them are i. acute? ii. obtuse? iii. right?

8. An angle is increased by 45°. Its complementary angle will be i. greater than 45° or ii. equal to 45° or iii. less than 45°?

9. Fill in the blanks.

i. If two angles are complementary, then their addition is ___________________.

ii. If two angles are supplementary, then their addition is ______________.

iii. If two adjacent angles are supplementary, they form a ______________.

II. PAIRS OF LINES

1. Give the properties that are utilized in each of the following.

i. If a || b, then ∠1 = ∠5.

ii. If ∠4 = ∠6, then a || b.

iii. If ∠4 + ∠5 = 180°, then a

2. From the image given below, find the, i. pairs of corresponding angles.

ii. pairs of alternate interior angles.

iii. pairs of interior angles on the same side of the transversal.

iv. vertically opposite angles.

3. In the adjoining figure, p || q. Find the unknown angles.

4. Find the value of x in each of the following figures if l || m.

5. In the given figure, the arms of two angles are parallel. If ∠ABC = 70°, then find i. ∠DGC

∠DEF

6. In the given figures below, decide whether l is parallel to m.

WORKSHEET - 2

I. MULTIPLE CHOICE QUESTIONS WITH SINGLE CORRECT ANSWER

1. In the given figure, the value of x is

2. In the adjoining figure, the value of ∠AOC such that AOB is a line is (2x - 25o) (3x + 5o)

a. 40° b. 55° c. 125° d. 180°

3. The supplement of an acute angle is: a. Acute b. Obtuse c. Right d. Straight

4. In the figure, PQ is a

a. Line b. Ray c. Line segment d. Ray or line

5. In the figure, find the value of x and y respectively.

a. 60°, 120° b. 70°, 110° c. 50°, 130° d. 80°, 100°

6. In the figure, if ∠ AOB is a straight line, then find the value of x.

7. In the given figure, if ∠POR = 90° and OQ bisects ∠POS, then find the value of 2y + z

8. Find two supplementary angles, if angles are in the ratio 7:11.

70°, 120°

60°, 120°

9. In the following figure, find the value of x

10. What is the value of x in the given figure?

11. Find the value of x in the figure given below.

12. An angle is 45° less than two times of its supplement, then find the greater angle.

75°

100°

120°

13. Two angles are complementary and equal, then find each of the angles.

90°, 90°

45°, 45°

14. The angle between the two hands of a clock at 4:30 p.m. is

45°

90°

15. If an angle is its own supplementary angle, then its measure is

30°

45°

16. Find ∠ AOC, if ∠ BOC = 60° .

17. In the figure, ABCD is a trapezium. FG is a straight line. Find ∠EBF.

105°

18. Which of the following is false?

19. In the figure, ∠CBD is a right angle. ∠DBE is thrice of ∠ABE. Find ∠ABC.

142° b. 124° c. 147° d. 174°

20. An angle is greater than 50°. Its complementary angle is

a. Greater than 40°

c. Equal to 40°

b. Less than 40°

d. Less than or equal to 40°

21. In the given figure, p || q || r and l || m. Find the value of b + c.

22. In the given figure p || q. What is the measure of x?

165°

175°

23. In the given figure, l || m. Find the measure of θ .

100°

80°

24. Two corresponding angles, A and B, are equal to 4y - 13° and y + 23°, respectively. What is the measure of each corresponding angle? a. 123°

57°

34°

35°

25. In the following figure, lines l and m are parallel to each other. t and w are two transversals intersecting the lines at A and B, and C and D. Find the values of x and y, respectively.

95°, 85°

75°, 85° 26. Given, AB || ED, AG || CB and F ⊥ AB, ∠FAG = 48° , ∠CDE = 37°. Find the value of x.

27. Find y, if 2x + z = 3y.

80°

II. MATCH THE FOLLOWING

List-I

30°

List-II

P. Two angles whose sum is 90° are called 1. Adjacent angles

Q. Two angles whose sum is 180° are called 2. Complementary angles

R. Two angles that are formed by two intersecting lines, angles which are not adjacent are called 3. Supplementary angles

S. Two angles with a common vertex, a common arm and the other arms lying on the opposite sides of the common arm form a pair of

a. P-1, Q-2, R-3, S-4

c. P-2, Q-3, R-4, S-1

List-I

4. Vertically opposite angles

b. P-3, Q-1, R-4, S-2

d. P-4, Q-2, R-1, S-3

List-II

P. If the measure of two supplement angles are (3x + 15°) and (2x + 5°), then the value of x is 1. 62°

Q. The complement angle of ) 2 5 of 70° is 2. 32°

R. Two complementary angles are in the ratio 7:8. The largest angle is 3. 48°

S. The supplement angle of 3 4 of 160° is 4. 60°

a. P-2, Q-1, R-3, S-4

c. P-2, Q-3, R-4, S-1

b. P-3, Q-1, R-4, S-2

d. P-4, Q-2, R-1, S-3

III. ASSERTION AND REASON

1. Assertion: If two lines intersect, then the vertically opposite angles are equal.

Reason: The sum of all the angles around a point is 180° .

a. Both assertion and reason are true and reason is the correct explanation of assertion.

b. Both assertion and reason are true but reason is not the correct explanation of assertion.

c. Assertion is true but reason is false.

d. Assertion is false but reason is true.

2. Assertion: The complement of 35° is 55°

Reason: If two angles are complementary, then their sum is 90° .

a. Both assertion and reason are true and reason is the correct explanation of assertion.

b. Both assertion and reason are true but reason is not the correct explanation of assertion.

c. Assertion is true but reason is false.

d. Assertion is false but reason is true.

3. Assertion: A line AB is denoted by AB  .

Reason: A line has no endpoints.

a. Both assertion and reason are true and reason is the correct explanation of assertion.

b. Both assertion and reason are true but reason is not the correct explanation of assertion.

c. Assertion is true but reason is false.

d. Assertion is false but reason is true.

4. Assertion: The supplement of 1 2 of 120° is 60° .

Reason: If two angles are supplementary, then their sum is 180°

a. Both assertion and reason are true and reason is the correct explanation of assertion.

b. Both assertion and reason are true but reason is not the correct explanation of assertion.

c. Assertion is true but reason is false.

d. Assertion is false but reason is true.

5. Assertion: Two parallel lines do not intersect each other.

Reason: The distance between two parallel lines is the same everywhere, i.e., it neither increases nor decreases.

a. Both assertion and reason are true and reason is the correct explanation of assertion.

b. Both assertion and reason are true but reason is not the correct explanation of assertion.

c. Assertion is true but reason is false.

d. Assertion is false but reason is true.

IV. SUBJECTIVE QUESTIONS

1. Prove that two lines which are parallel to the same given line are parallel to each other.

2. In the given figure, two straight lines, AB and CD, intersect at a point O. If ∠AOC = 42°, then find the measure of each of the angles given below.

i. ∠AOD ii. ∠BOD iii. ∠COB

3. An angle is 45° less than two times its supplement. Find the angles.

4. In the given figure, AOB is a straight line and the ray OC stands on it.

Find the value of x. Also, find ∠AOC and ∠BOC. (3x - 20o) (2x - 10o)

5. In the given figure, EF || GH, ∠EAB = 65° and ∠ACH = 100°. Determine:

∠BAC

6. In the given figure, show that CD || EF.

LINES AND ANGLES

7. In the given figure, l || m and t is a transversal such that ∠1 = 135°

Find the measure of each of the angles ∠2, ∠3, ∠4, ∠5, ∠6, ∠7 and ∠8. l t 4 1 2 3 8 5

8. Find the complement of each of the following angles: i. 60° ii. 25° iii. 72°

9. Find the supplement of each of the following angles: i. 125° ii. 64° iii. 38°

10. In the adjoining figure, AB || CD and EF is a transversal which intersects them at G and H, respectively. If GL and HM are the bisectors of the corresponding angles EGB and EHD, respectively, show that GL || HM.

ANSWER KEY

1: INTEGERS

Worksheet 1

I. Basics of integers, Properties of addition and subtraction of integers

1. i. 4 ii.–30

2. i. –304 ii.–31

3. i. –82 ii.534

4. i. –268 ii.39

5. –56

6.158

7. ( 1, 19) 8.83

9. NA

10. Temperature on Friday = –2°C Temperature on Saturday = 13°C

II. Multiplication of Integers, Properties of multiplication of Integers

1. i. 84 ii.–120

2. i. 48 ii.–180

3. i. –32 ii.–1

4. Negative

5. Positive

6. i. 60 ii.70

7. 6

8. NA

9.8°C

10. 15 points

III. Division of Integers, Properties of division of Integers

1. i. –5 ii.12 iii.–1

2. i. –41 ii.0 iii.–8

3. –128

4. i. –104 ii.–18

5. 80 mins or 1 hr 20 mins

6. 15 mins

7. –15

8.161

9. 6 board games

10. 10 cookies

Worksheet 2

I. Multiple choice questions with single correct answer

1. b 2. d 3. a 4. d 5. C 6. d 7. b 8. a 9. c 10. b

11. c 12. d 13. c 14. a 15. b 16. a 17. b 18. a 19. d 20. a

II. Fill in the blanks 1.22

2. 1 3. –10

4. 0 or –6

5. Negative 6. 6 7.16

8. –13

9. –60

10.16°C

III. Subjective questions

1. –31 is 33 less than 2.

2. NA

3. –599

4. ( 14, 2)

5. i. –20 ii.225

6. –105

7. 3°C

8. 30 mins

9. 0

10. –5

2: FRACTIONS AND DECIMALS

Worksheet 1

I. Fractions, Addition, Subtraction, Multiplication, and Division of fractions

1. a - ii, b - i, c – iii

2. a - ii, b - iii, c - i

3. 5 8

4. 1 8

5. a. i. 1 ii.23 b.i. 12 ii.18

ANSWER KEY

c.i. 12 ii.27

d.i. 16 ii.28

e.i. 5 3 8 ii. 2 1 9

f.i. 2 19 48 ii. 6 1 24

6. a. 15 3 5 b. 33 3 4

c. 15 3 4 d. 25 1 3

e. 19 1 4 f.27 1 5

7. i. 2 litres ii. 3 5

8. i. 3 5 of 5 8 is greater. ii. 1 2 of 6 7 is greater.

9. 2 1 4 m

10. 10 1 2 m

11.44km

12. i. 16 ii. 84 5

iii. 24 7 iv. 9 7

v. 7 5

13. i. 3 7 ; Reciprocal = 7 3. It is an improper fraction.

ii. 5 8 ; Reciprocal = 8 5. It is an improper fraction,

iii. 9 7 ; Reciprocal = 7 9 . It is a proper fraction.

iv. 6 5 ; Reciprocal = 5 6, It is a proper fraction.

v. 12 7 ; Reciprocal = 7 12. It is a proper fraction.

vi. 1 8 ; Reciprocal = 8 1 . It is a whole number

vii. 1 11 ; Reciprocal = 11 1 . It is a whole number

14. 5 1 3 m

15.300

II. Decimals, multiplication, and division of decimals

1. i. 1.2

ii. 36.8

iii. 13.55 iv.80.4

v.0.35 vi.844.08

vii.1.725

2. 17.1 cm2

3. i. 13 ii.368

iii.1537 iv.1680.7

v.3110 vi.15610

vii.362 viii.4307

ix.5 x.0.8 xi.90 xii.30

4. 553 km

5. i. 0.75 ii.5.17

iii.63.36 iv.4.03

v.0.025 vi.1.68

vii.0.0214 viii.10.5525

ix.1.0101 x.110.011

6. i. 0.48 ii.5.25

iii.0.07 iv.3.31

v.27.223vi.0.056

vii.0.397

7. i. 0.4027 ii.0.003

iii.0.0078iv.4.326

v.0.236 vi.0.9853

8. i. 0.0079 ii.0.03853

iii.0.1289iv.0.0005

9. i. 2 ii.180

iii.6.5 iv.44.2

v.2 vi.31

vii.510 viii.27

ix.2.1

10.18 km

Worksheet 2

I. Multiple choice questions with single correct answer

1.a 2. d 3. c 4. c 5. b

6.a 7. b 8. a 9. d 10. d

11.c 12. b 13. b 14. b 15. a

16.c 17. a 18. a 19.c 20. c

21.d 22. b 23. b 24. a 25. d

II. Subjective questions

1. i. 1 5 ii.3 iii. 9 7 iv. 5 21

2. 125 52 3.121.296

4. ₹ 35

5.2.304

6. ₹ 3000

7.25

8. 6 kg

9. 2 1 5

ANSWER KEY

Worksheet 2

I. Multiple choice questions with single correct answer

II. Fill in the blanks

1. 4.5

III. Case study 4. c 5. d

10. 40.5 km 1. d 2. b 3.b

3: DATA HANDLING

Worksheet 1

I. Introduction to data handling

1. NA

2. i. 9 ii. 1 iii. 8 iv.4.5

3. 27 cm

4.56

II. Measures of central tendency

1. 170 cm

2. 40.5 kg

3. 8 4.23

5. 4 6.16 7.19 8.80

III. Bar Graph, Double Bar Graph

1. a. The production of cement during the years 2003-08.

b.250 lakh tonnes

c.2004-05

2. a. Expenditure on education in four countries.

b. Expenditure on education in 1990 is maximum in Australia.

c. The expenditure has gone up from 1990 to 2000 in East Africa and Latin America.

3. NA

4. NA

2. 42

3. 50 kg

4. 5

5. 125 cm

III. Subjective questions

1. 5.6

2. 7

3. 9.5

4. NA

5. NA

4: SIMPLE EQUATIONS

Worksheet 1

I. Solving an equation

1. i. �� = 2 5 ii. �� =4 iii. �� =3 iv. �� =1 v. �� = 2 vi. �� = 2 3

II.Solving an equation using transposition and variable on both sides

1. �� = 10

2. �� = 3. �� = 15 4. i. �� = 11 4 ii.�� =6

5. �� =2

6. �� =1

7. �� =1

8. 20 9. �� =1

10. �� = 25

III.Applications of simple equations

1. 19 & 11

2. 3 and 4 �� = 9 1 12

ANSWER KEY

3. 35 & 7

4. 36

5. 62 or 26

6. 5 yrs. & 30 yrs.

7. 6 yrs. & 60 yrs.

8. 21 yrs.

9. A = 30 yrs., B = 60 yrs.

10. 7

11. 31 notebooks

12. 7, 8, 9

13. ₹ 85.

14. 8

15. 23 kg

16. 20

17. L = 262 m, B = 255 m

18. 16, 48

19. David = 73 runs, Rajat = 219 runs

20. 40

21. 70°.

22. Rahul = 66 runs, Sachin = 132 runs

Worksheet 2

I. Multiple choice questions with single correct answer

1. a 2. b 3. b 4. c 5. d

6.b 7. a 8. b 9. d 10.c

11.c 12. d 13.b 14. b 15. b

16.d 17. b 18. c 19. c 20. a

21. c 22. d 23. b 24. a 25. d

26.c 27. b

II. Fill in the blanks

1.42

2.13,10

3.29

4. 2

5. 5 cm

6. ��

7. 8 5

8. 11

III.Subjective questions

1. i. 5�� = 30

ii. �� 7=4

iii. �� + 10 = 15

iv. 3�� 5= 27

v. 20 −�� = 15

vi. �� 5= 12

2. x = 10

3. y = 6

4. x = 8

5. x = 17.6

6. Each side = 22 units, Perimeter = 55 units

sides: 22 cm, 22 cm, 11 cm perimeter: 55 cm

7. Width = 12 m, Length = 36 m

8. 250 coins

9. 5 yrs

10. 17 yrs

11. x = 5

5: LINES AND ANGLES

Worksheet 1

I. Introduction and related angles

1. i. 70° ii.27°iii.33°

2. i.75° ii.93° iii.26°

3. i. Supplementary ii.Complementary iii.Supplementary iv.Supplementary v.Complementary vi.Complementary

4.45o

5.90o

6. ∠2 will be increased by same quantity.

7. i. No ii. No iii. Yes

8. iii

9. i. 90o ii.180o

iii.Linear Pair

II. Pairs of lines

1. i. Corresponding angles are equal. ii.Alternate interior angles are equal. iii.Co-interior angles are supplementary.

2. i. ∠4 and ∠8, ∠1 and ∠5

∠3 and ∠7, ∠2 and ∠6

ii.∠3 and ∠5, ∠2 and ∠5

iii. ∠3 and ∠8, ∠2 and ∠5

iv. ∠1 and ∠3, ∠2 and ∠4

∠6 and ∠8, ∠5 and ∠7

ANSWER KEY

3. f = 55o, e = 55o, a = 55o, d = 125o, c = 55o, b = 125o

4. i. 70o

ii.100o

5. i. 70o ii. 70o

6. i. No ii.No iii.Yes iv.No

Worksheet 2

I. Multiple choice questions with single correct answer

1. a 2. c 3. b 4. c 5. a

6. c 7. c 8. c 9. d 10. c

11. a 12. d 13. b 14. a 15. d

16. a 17. b 18. c 19. d 20. b

21. a 22. c 23. b 24. d 25. c

26. c 27. d

II. Match the following

1. c 2 a

III. Assertion and reason

1. c 2. a 3 d 4. d 5. a

IV. Subjective questions

1. NA

2. i. 138o ii. 42o iii. 138o

3. 105o and 75o

4. x = 42o , ∠AOC = 74o, ∠BOC = 106o ii. 80o

5. i. 35o

6.NA

7.∠2 = 45o, ∠3 = 135o, ∠4 = 45o, ∠5 = 45o, ∠6 = 135o, ∠7 = 45o, ∠8 = 135o

8. i. 120o ii. 155o iii. 108o

9. i. 55o ii. 116o iii. 142o

10. NA