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In class VI, we have learnt about addition and subtraction of integers. In this lesson, we shall learnvarious properties of integers. These include closure, commutative, associative and distributiveproperty. Existence of identity element and inverse of an element will also be discussed. We shallalso deal with multiplication and division of integers.

Let’s review what we have learnt in our previous class.

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The numbers 1, 2, 3, 4, … are called natural numbers.

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The numbers 0, 1, 2, 3, 4, … are called whole numbers.

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The numbers …, –4, –3, –2, –1, 0, 1 , 2 , 3, 4, … are called integers.

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The number 0 is an integer which is neither positive nor negative.

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Modulus or absolute value of an integer is always positive. It represents the numerical value of aninteger (i.e., ignore sign). The absolute value of an integer a is denoted by |a| and is given by

if 0 if 0

a aaa a

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We represent integers on a number line as:

We have learnt addition and subtraction of integers using a number line in the previous class.

We know that on a number line, when we:

(i) add a positive integer or subtract a negative integer we move to the right.

(ii) subtract a positive integer or add a negative integer we move to the left.

Example 1: The following number line shows the temperature in degree Celsius (°C) at differentplaces on a particular day.

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... –4 –3 –2 –1 0 1 2 3 4 ...

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(a) Observe the number line and write the temperature of the places marked on it.(b) What is the temperature difference between the hottest and the coldest places

shown above?(c) Temperature of which place lies between Srinagar and Ooty?(d) Can we say the difference in temperature of Mysore and Ooty is less than the

difference in temperature of Ooty and Goa?

Solution: (a) Temperature of the places marked on the number line are as follows:(i) Srinagar = –3°C (ii) Mysore = 6°C (iii) Ooty = 11°C

(iv) Goa = 15°C (v) Delhi = 24°C(b) Difference in temperature of two extreme climates = 24° – (–3°) = 27°C(c) Temperature of Mysore lies between Srinagar and Ooty.(d) Difference in temperature of Mysore and Ooty = 11° – 6° = 5°C

Difference in temperature of Ooty and Goa = 15° – 11° = 4°CThus, difference in temperature of Mysore and Ooty is not less than that of Ootyand Goa.

Example 2: In a quiz, positive marks are given for correct answers and negative marks are givenfor incorrect answers. If Sam’s scores in five successive rounds were 30, –5, –10, 20and 5, what was his total after five rounds?

Solution: Total marks scored by Sam in five successive rounds= 30 + (–5) + (–10) + 20 + 5 = (30 + 20 + 5) – (5 + 10)= 55 – 15 = 40

Example 3: In Mussoorie, the temperature was 4°C on Monday and then it dropped by 6°C onTuesday. What was the temperature of Mussoorie on Tuesday? On Wednesday, it roseby 4°C. What was the temperature on this day?

Solution: Temperature of Mussoorie on Monday = 4°C

Temperature on Tuesday = 4°C – 6°C = –2°C (� it dropped by 6°C)

Temperature on Wednesday = –2°C + 4°C = 2°C (� it rose by 4°C)

Example 4: A plane is flying at a height of 4,000 m above the sea level. At a particular point, itis exactly above a submarine floating 1,000 m below the sea level. What is the verticaldistance between them?

Solution: The height of the plane = +4,000 mThe depth of the submarine = –1,000 m (� sea level is denoted by 0)Vertical distance between the plane and submarine = 4,000 – (–1,000) = 5,000 m

Example 5: Mansi deposits � 5,000 in her bank account and withdraws � 3,500 from it, the nextday. If amount withdrawn from the account is represented by a negative integer, thenhow will you represent the amount deposited? Find the balance in Mansi’s accountafter the withdrawal.

Srinagar Mysore Ooty Delhi

–5 0 5 10 15 20 25

Goa

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Solution: The amount deposited will be represented by a positive integer.Amount deposited = + � 5,000

Amount withdrawn = – � 3,500Balance in Mansi’s account = � 5,000 – � 3,500 = � 1,500

Example 6: Vivek and James had a boxing match which extended to five bouts. In the first boutVivek gained 5 points. In subsequent bouts, he lost 7 points, gained 4 points, lost 6points and finally gained 3 points over his opponent James. Who won the match?

Solution: Vivek’s total points in the 5 bouts = +5 – 7 + 4 – 6 + 3= (5 + 4 + 3) – (7 + 6)= 12 – 13 = –1

Since, Vivek ended with one negative point, therefore James won the match.

Example 7: Verify a – (–b) = a + b for the following values of a and b.(a) a = 30, b = 12 (b) a = 65, b = 75

Solution: (a) a = 30, b = 12We have, a – (–b) = 30 – (–12) = 30 + 12 = 42Also, a + b = 30 + 12 = 42; hence, a – (–b) = a + b

(b) a = 65, b = 75We have, a – (–b) = 65 – (–75) = 65 + 75 = 140Also, a + b = 65 + 75 = 140; hence, a – (–b) = a + b

Example 8: Use the sign of >, < or = in the box to make the statements true.

(a) (–7) + (–5) (–7) – (–5)

(b) 50 – 60 + 20 25 – 60 – 5

(c) –2 + 13 – (–7) 15 – (–6) + (–3)

Solution: (a) � (–7) + (–5) = –(7 + 5) = –12 and (–7) – (–5) = –7 + 5 = –(7 – 5) = –2

∴ (–7) + (–5) (–7) – (–5)

(b) � 50 – 60 + 20 = (50 + 20) – 60 = 70 – 60 = 10 and25 – 60 – 5 = 25 – (60 + 5) = 25 – 65 = –(65 – 25) = –40

∴ 50 – 60 + 20 25 – 60 – 5

(c) � –2 + 13 – (–7) = (13 + 7) – 2 = 20 – 2 = 18 and15 – (–6) + (–3) = (15 + 6) – 3 = 21 – 3 = 18

∴ –2 + 13 – (–7) 15 – (–6) + (–3)

Example 9: A water tank has steps inside it. A monkey is sitting on the topmost step (i.e., the firststep). The water level is at the ninth step.(a) He jumps 3 steps down and then jumps back 2 steps up. In how many jumps will

he reach the water level?(b) After drinking water, he wants to go back. For this, he jumps 4 steps up and then

jumps back 2 steps down in every move. In how many jumps will he reach backthe top step?

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(c) If the number of steps moved down is represented by negative integers and thenumber of steps moved up by positive integers, represent his moves in part (a)and (b) by completing the following:(i) –3 + 2 – ____ = –8 (ii) 4 – 2 + ____ = 8

If in (i) the sum (–8) represents going down by eight steps, what will the sum 8in (ii) represent? (NCERT)

Solution: (a) Let the number of steps moved down be represented by negative integers and thenumber of steps moved up by positive integers. Therefore, the movement ofmonkey can be represented as –3 + 2 – 3 + 2 – 3 + 2 – 3 + 2 – 3 + 2 – 3 = –8, i.e.,he has to jump 11 times to reach the water level.

(b) Number of jumps required to reach back can be calculated as:4 – 2 + 4 – 2 + 4 = 8i.e., he has to jump 5 times to reach back.

(c) The moves in (i) and (ii) have been shown in (a) and (b) above. The sum 8 in (ii)represents going up by 8 steps.

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1. Find the sum of the following integers:(a) 8 and 21 (b) 25 and –13 (c) –16 and –8 (d) 35 and –11

2. Evaluate the following either by direct method or by using a number line:(a) 3 + (–2) (b) –3 + 4 + (–7) (c) 8 – (–2) – 3 (d) 7 + (–5) + 1

3. Represent the following as integers with appropriate sign:(a) Temperature of Leh is 13ºC below 0ºC. (b) A well is 130 m deep.(c) The height of TV tower is 90 m. (d) A gain of � 50,000

4. Complete the following table:

+ –6 –3 –2 –1 0 2 5 7

1

3

8

–1

–5

5. Arrange the following integers in descending order:(a) –45, 34, –890, 219 (b) 678, –980, 432, –675

6. The midnight temperature of Delhi in the month of December was 10ºC. It dropped by 4ºC inthe next three hours. What was the temperature after 3 hours?

7. Morning temperature on a particular day in Chandigarh in the month of October was recordedas 13.5ºC. However, by afternoon it rose by 7ºC. What was the temperature recorded in theafternoon?

8. A plane is flying at a height of 5,850 m. A submarine is floating at a depth of 700 m below thesea level exactly below the plane at a particular time. What is the distance between the planeand the submarine?

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9. In a quiz, 2 positive marks are allotted for each correct answer and –1 for eachincorrect answer. In four rounds of quiz, Reshma scored 45, –34, 40 and –36marks and Radha scored –42, –12, 28 and 38 marks. Who scored more marks?

10. A monkey is trying to climb a pole measuring 15 m, which is greased. He triesto climb 3 m and slips down 2 m every time. In how many attempts will themonkey be able to climb 10 m of the pole?

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ClosureThe sum of any two integers is again an integer.Since 3 and 5 are integers, ∴ 3 + 5 = 8 is also an integer.Since 2 and –4 are integers, ∴ 2 + (–4) = 2 – 4 = –2 is again an integer.

CommutativeIf a and b are any two integers, then a + b = b + a.Since 3 and 6 are integers, ∴ 3 + 6 = 9 = 6 + 3.Since –3 and 5 are integers, ∴ (–3) + 5 = 2 = 5 + (–3).

AssociativeIf a, b and c are any three integers, then (a + b) + c = a + (b + c).Since 2, 3 and 5 are integers, ∴ (2 + 3) + 5 = 10 = 2 + (3 + 5).Since 2, –3 and 5 are integers, ∴ {2 + (–3)} + 5 = 4 = 2 + (–3 + 5).

Existence of Additive IdentityIf a is any integer, then a + 0 = a = 0 + a. We say 0 is an additive identity for integers.Since 5 is an integer, ∴ 5 + 0 = 5 = 0 + 5.Since –5 is an integer, ∴ (–5) + 0 = –5 = 0 + (–5).Hence 0 is the additive identity for integers.

Existence of Additive InverseIf a is any integer, then a + (–a) = 0 = (–a) + a. We say –a is an additive inverse of a.Since 6 + (–6) = 0 = (–6) + 6, ∴ additive inverse of 6 is –6.

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ClosureIf we subtract any two integers the result is again an integer.Since 5 and 3 are integers, ∴ 5 – 3 = 2 and 3 – 5 = –2 are also integers.Since –2 and 4 are integers, ∴ –2 – 4 = –6 is an integer. Also, 4 – (–2) = 6 is also an integer.

Not CommutativeIf a and b are any two integers, then a – b ≠≠≠≠≠ b – a, i.e., commutative property does not hold forsubtraction of integers.For integers –3 and 5 we have –3 – (+5) = –(3 + 5) = –8 and 5 – (–3) = 5 + 3 = 8,

∴ –3 – (+5) ≠ 5 – (–3)Not AssociativeIf a, b and c are integers, then (a – b) – c ≠ a – (b – c). Associative property does not hold good forsubtraction of integers.

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For integers –1, +6 and 7 we have (–1 – 6) – 7 = –7 – 7 = –14

and –1 – (6 – 7) = –1 – (–1) = –1 + 1 = 0

∴ (–1 – 6) – 7 ≠ –1 – (6 – 7)

Example 10: Write down a pair of integers whose:(a) sum is –4 (b) difference is –10 (c) sum is 0

Solution: One such pair could be(a) –7 and 3 (b) 3 and 13 (c) –2 and 2

Note: There are infinite pairs of integers satisfying the above cases.

Example 11: (a) Write a pair of negative integers whose difference is 7.(b) Write a negative integer and a positive integer whose sum is –5.(c) Write a negative integer and a positive integer whose difference is –5.

Solution: (a) –3 and –10, � –3 – (–10) = –3 + 10 = 7.(b) –8 and 3, � –8 + 3 = –(8 – 3) = –5.(c) –4 and +1, � –4 – (+1) = –4 – 1 = –5.

Note: There are infinite pairs of integers satisfying the above cases.

Example 12: In a quiz, team A scored –40, 50, 0 and team B scored 50, 0, –40 in three successiverounds. Which team scored more? Can integers be added in any order?

Solution: Total score of team ‘A’ = –40 + 50 + 0 = (–40 + 50) + 0 = 10 + 0 = 10Total score of team ‘B’ = 50 + 0 – 40 = 50 + {0 + (–40)} = 50 – 40 = 10Thus, the score of both the teams is same.Yes, we can add integers in any order.

Example 13: Fill in the blanks to make the following statements true:(a) (–3) + (–7) = (–7) + (...........) (b) –28 + ........... = 0(c) 19 + (...........) = 0 (d) {9 + (–12)} + (...........) = 9 + {(–12) + (–7)}

Solution: (a) –3 (b) 28 (c) –19 (d) –7

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1. Write down a pair of integers whose:

(a) sum is –9 (b) difference is –18 (c) sum is –12. Write a negative integer and a positive integer whose sum is –15.3. Write a negative integer and a positive integer whose difference is 11.4. In a quiz, team A scored 20, –10, 34 and team B scored –10, 20, 34 in three successive rounds.

Which team scored more?

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We all know how to multiply natural numbers. Now we will learn how to multiply two integers. Infact multiplication of integers is equal to repeated addition. We will use number line to understandmultiplication of integers.

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Case I: Multiplication of two positive integers, say +2 and +3Step 1: Let’s move two steps to the right from 0 to reach at 2.

Step 2: On making a total of three such moves to the right we reach at 6.

Thus, the product of 2 and 3 is 6, i.e., (+2) × (+3) = +6.

Case II: Multiplication of one negative and one positive integer, say –3 and 2Step 1: Let’s move three steps to the left from 0 to reach at –3.

Step 2: On making a total of two such moves we reach at –6.

Thus, the product of (–3) and 2 is (–6), i.e., (–3) × (2) = –6.

Case III: Multiplication of two negative integers, say –3 and –2We may write (–3) × (–2) as (–3) × 2 × (–1).Let’s first represent (–3) × 2:

Step 1: Let’s move three steps to the left from 0 to reach at –3.

Step 2: On making two such moves we reach at –6.

Thus, (–3) × 2 = –6.Since, (–3) × (–2) = (–3) × 2 × (–1) = (–6) × (–1) = –(–6)

Therefore, the product of –3 and –2 is 6.

• A positive integer multiplied by another positive integer gives a positive integer.• A positive integer multiplied by a negative integer gives a negative integer.• A negative integer multiplied by another negative integer gives a positive integer.

–2 –1 0 1 2 3 4 5 6 7

–2 –1 0 1 2 3 4 5 6 7

–7 –6 –5 –4 –3 –2 –1 0 1 2

–7 –6 –5 –4 –3 –2 –1 0 1 2

–7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7

Since, –(–6) = 6

–7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7

–7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7

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ClosureThe product of any two integers is again an integer.For integers –2 and 4, we have –2 × 4 = –8 which is again an integer.

CommutativeIf a and b are any two integers, then a × b = b × a.For integers 3 and –7, we have 3 × (–7) = –21 = –7 × 3.

AssociativeIf a, b and c are any three integers, then (a × b) × c = a × (b × c).For integers 2, –3 and –4, we have {2 × (–3)} × (–4) = –6 × (–4) = 24 and 2 × {–3 × (–4)} =2 × 12 = 24. Hence, {2 × (–3)} × (–4) = 2 × {–3 × (–4)}.

DistributiveIf a, b and c are any three integers, then a × (b + c) = (a × b) + (a × c).For integers –7, –3 and 2, we have –7 × (–3 + 2) = –7 × (–1) = 7and –7 × (–3) + (–7) × 2 = 21 – 14 = 7.Hence, –7 × (–3 + 2) = –7 × (–3) + (–7) × 2

Existence of Multiplicative IdentityIf a is any integer, then a × 1 = a = 1 × a. The number 1 is known as the multiplicative identityof a.For integer 4, we have 4 × 1 = 4 = 1 × 4 and for integer –3, we have –3 × 1 = –3 = 1 × –3.Hence, 1 is the multiplicative identity for integers.

Property of ZeroFor any integer a, we have (a × 0) = 0 = (0 × a).For integer –3, we have –3 × 0 = 0 = 0 × –3 and for integer 4, we have 4 × 0 = 0 = 0 × 4.

Example 14: Find each of the following products:(a) (–1) × 227 (b) (–416) × (–1) (c) (–13) × 0 × (–18)(d) (–10) × (–5) × (–4) (e) (–3) × (–5) × (–2) × (–1)

Solution: (a) (–1) × 227 = –227 (b) (–416) × (–1) = 416(c) –13 × 0 × (–18) = (–13 × 0) × (–18) = 0 × –18 = 0(d) (–10) × (–5) × (–4) = {(–10) × (–5)} × (–4) = 50 × (–4) = –200(e) (–3) × (–5) × (–2) × (–1) = {(–3) × (–5)} × {(–2) × (–1)} = 15 × 2 = 30

Example 15: Verify the following:(a) 9 × {7 + (–3)} = {9 × 7} + {9 × (–3)}(b) (–20) × {(–4) + (–6)} = {(–20) × (–4)} + {(–20) × (–6)}

Solution: (a) 9 × {7 + (–3)} = {9 × 7} + {9 × (–3)}LHS = 9 × {7 + (–3)} = 9 × {7 – 3} = 9 × 4 = 36RHS = {9 × 7} + {9 × (–3)} = 63 + (–27) = 63 – 27 = 36� LHS = RHS, hence verified.

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(b) (–20) × {(–4) + (–6)} = {(–20) × (–4)} + {(–20) × (–6)}LHS = (–20) × {(–4) + (–6)} = (–20) × (–10) = 200RHS = {(–20) × (–4)} + {(–20) × (–6)} = 80 + 120 = 200� LHS = RHS, hence verified.

Example 16: (a) For any integer n, what is (–1) × n equal to?(b) Determine the integer whose product with (–1) is:

(i) –21 (ii) 31 (iii) 0Solution: (a) For any integer n, (–1) × n = –n.

(b) Integer whose product with (–1) is:(i) –21 is 21 (ii) 31 is –31 (iii) 0 is 0

Example 17: Find the value of the expression ( |–12| + |2 – 26|) × |13 – 25|

Solution: ( |–12| + |2 – 26|) × |13 – 25| = (12 + |–24|) × |–12| = (12 + 24) × 12= 36 × 12 = 432

Example 18: Evaluate the following using suitable properties. Also mention the properties used.(a) (275 × 163) × 0 (b) 325 × (–23) + 325 × 23

Solution: (a) (275 × 163) × 0 = 274 × (163 × 0) (Associative property)= 275 × 0 = 0

(b) 325 × (–23) + 325 × 23 = 325 × (–23 + 23) (Distributive property)= 325 × 0 = 0

Example 19: Find the product using suitable properties:(a) 26 × (–37) + (–37) × 74 (b) 8 × 47 × (–125)(c) (–17) × 102 (d) (–625) × (–47) + (–625) × (–53)

Solution: (a) 26 × (–37) + (–37) × 74 = (–37) × 26 + (–37) × 74 (Commutative property)= (–37) × (26 + 74) (Distributive property)= (–37) × 100 = –3,700

(b) 8 × 47 × (–125) = 47 × 8 × (–125) (Commutative property)= 47 × {8 × (–125)} (Associative property)= 47 × –1,000 = –47,000

(c) (–17) × 102 = (–17) × (100 + 2)= (–17) × 100 + (–17) × 2 (Distributive property)= –1,700 – 34 = –1,734

(d) (–625) × (–47) + (–625) × (–53) = (–625) × {–47 + (–53)}(Distributive property)

= (–625) × (–100) = 62,500

Example 20: A certain freezing process requires the room temperature to be lowered from 30°C atthe rate of 5°C every hour. What will be the room temperature 8 hours after theprocess begins?

Solution: Temperature to be lowered by 5°C every hour.∴ Temperature to be lowered in 8 hours = 8 × 5°C = 40°C

Room temperature in the beginning = 30°C∴ Room temperature after 8 hours = 30°C – 40°C = –10°C.

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Example 21: In a class test containing 20 questions, 4 marks are awarded for every correct answer,1 mark is deducted for every incorrect answer and 0 is given for questions notattempted. Answer the following questions:

(a) Rohit gets four correct and six incorrect answers. What is his score?

(b) Arushi gets five correct and five incorrect answers. What is her score?

Solution: (a) Rohit’s score = 4 × 4 + 6 × (–1) = 16 – 6 = 10(b) Arushi’s score = 5 × 4 + 5 × (–1) = 20 – 5 = 15

Example 22: A cement company earns a profit of � 8 per bag of white cement sold and a loss of� 5 per bag of grey cement sold.

(a) The company sells 3,000 bags of white cement and 5,000 bags of grey cement ina month. What is its profit or loss?

(b) What is the number of white cement bags it must sell to have neither profit norloss, if the number of grey bags sold is 6,400? (NCERT)

Solution: (a) Profit or loss of the company = � 3,000 × 8 + 5,000 × (–5)

= � 24,000 – � 25,000 = � –1,000

∴ There is a net loss of � 1,000.

(b) Let the number of white cement bags sold for neither profit nor loss be n.

Now, 8 × n + (–5) × 6,400 = 0

⇒ 8n = 5 × 6,400

⇒ n =5 6,400

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⇒ n = 5 × 800 = 4,000

i.e., the company must sell 4,000 bags of white cement for no profit, no loss.

Example 23: Replace the blank with an integer to make it a true statement.

(a) (–5) × ________ = 20 (b) 8 × ________ = –32

(c) ________ × (–8) = –40 (d) ________ × (–7) = 84

Solution: (a) –4 (b) –4 (c) 5 (d) –12

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1. Multiply the following:(a) 12 and 7 (b) –8 and 8 (c) 9 and –11 (d) –6 and –5 (e) 0 and –45

2. Find each of the following products:(a) 2 × 3 × (–8) (b) (–5) × 7 × (–4) (c) (–6) × (–7) × (–9)(d) (–11) × (21) × 0 × (–34) (e) (–2) × (–5) × (–4) × (–10)

3. Simplify using suitable properties of multiplication of integers:(a) (–3) × 5 + (–3) × 3 (b) 7 × (–13) + 7 × (–10) (c) 10 × (–4) + 5 × (–4)(d) (–12) × (–7) + (–12) × (–3) (e) 16 × (–68) + 16 × (–32)

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4. In a competitive examination, 5 marks are awarded for each correct answer, 2 marks arededucted for each incorrect answer and no marks are given for a question which is not attempted.(a) Nishi appeared in the examination and attempted 12 questions correctly and 5 questions

incorrectly. However she did not attempt 3 questions. What will be her score?(b) Rohan attempted 7 questions correctly and 9 questions incorrectly. He did not attempt 4

questions. What will be his score?5. A company is producing two products A and B. The company makes a profit of � 52 per unit

on A and a loss of � 10 per unit on B. If the company sells 3,600 units of product A and 4,000units of product B, find the profit or loss of the company due to the sale.

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Division is inverse process of multiplication, i.e., 4 × 7 = 28 implies 28 ÷ 4 = 7 or 28 ÷ 7 = 4.Also, 5 × (–4) = –20 implies –20 ÷ 5 = –4 and –20 ÷ (–4) = 5.

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• When we divide two integers having the same sign (either both positive or both negative),we divide them regardless of their sign as whole numbers and then assign a positive signto the quotient.

• When we divide two integers having different signs (one positive and other negative), wedivide them regardless of their sign as whole numbers and then assign a negative sign tothe quotient.

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Closure Property Does Not Hold GoodThe closure property is not valid for integers. If a and b are two integers, then a ÷ b is notnecessarily an integer.

For example, 7 and 3 are integers but 7

3 is not an integer.

Not CommutativeIf a and b are two different integers, then (a ÷ b) ≠≠≠≠≠ (b ÷ a), i.e., division of integers is not commutative.

For integers 3 and 6, (3 ÷ 6) ≠ (6 ÷ 3).

Not AssociativeIf a, b and c are three integers such that b ≠ 0 and c ≠ 0, then (a ÷ b) ÷ c ≠≠≠≠≠ a ÷ (b ÷ c).The result is true only for c equal to 1.

Property of ZeroZero divided by any non-zero integer a is equal to zero, i.e., 0 ÷ a = 0 provided a ≠ 0.For example, 0 ÷ 5 = 0 and 0 ÷ (–7) = 0.

Caution: a ÷ 0 is meaningless, i.e., division by 0 is not defined.

Division by 1 Keeps the Integer UnchangedIf a is any integer, then a ÷ 1 = a, i.e., any integer divided by 1 remains as it is.

For integer 5, we have 5 ÷ 1 = 5 and for integer –3, we have –3 ÷ 1 = –3.

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Division of Integer by Itself

Any non-zero integer a, divided by itself gives 1 as the answer, i.e., a ÷÷÷÷÷ a =aa = 1.

For integer –2, we have –2 ÷ (–2) = 1 and for integer 3, we have 3 ÷ 3 = 1.

Example 24: Evaluate each of the following:(a) (–40) ÷ 10 (b) (–36) ÷ (–9) (c) (–39) ÷ (39)(d) 13 ÷ {(–3) + 2} (e) 0 ÷ (–8) (f) (–31) ÷ {(–30) + (–1)}(g) {(–36) ÷ 12} ÷ 3 (h) {(–7) + 5} ÷ {(–3) + 1}

Solution: (a) (–40) ÷ 10 = –4 (b) (–36) ÷ (–9) = 4 (c) (–39) ÷ (39) = –1(d) 13 ÷ {(–3) + 2} = 13 ÷ (–1) = –13 (e) 0 ÷ (–8) = 0(f) (–31) ÷ {(–30) + (–1)} = (–31) ÷ (–31) = 1(g) {(–36) ÷ 12} ÷ 3 = (–3) ÷ 3 = –1(h) {(–7) + 5} ÷ {(–3) + 1} = (–2) ÷ (–2) = 1

Example 25: Verify that a ÷ (b + c) ≠ (a ÷ b) + (a ÷ c) for each of the following values.(a) a = 12, b = 1, c = –2 (b) a = 10, b = 1, c = 1

Solution: To verify a ÷ (b + c) ≠ (a ÷ b) + (a ÷ c)(a) a = 12, b = 1, c = –2

LHS = a ÷ (b + c) = 12 ÷ {1 + (–2)} = 12 ÷ (–1) = –12

RHS = (a ÷ b) + (a ÷ c) = 12 12

1 2+

− = 12 – 6 = 6

� LHS ≠ RHS, hence verified.

(b) a = 10, b = 1, c = 1LHS = a ÷ (b + c) = 10 ÷ (1 + 1) = 10 ÷ 2 = 5RHS = (a ÷ b) + (a ÷ c) = (10 ÷ 1) + (10 ÷ 1) = 10 + 10 = 20� LHS ≠ RHS, hence verified.

Example 26: Fill in the blanks:(a) 469 ÷ _______ = 469 (b) (–80) ÷ _______ = –1 (c) (–205) ÷ _______ = 1(d) –89 ÷ _______ = 89 (e) _______ ÷ 1 = –83 (f) _______ ÷ 48 = –1(g) 30 ÷ _______ = –2 (h) _______ ÷ (3) = –4

Solution: (a) 469 ÷ 1 = 469 (b) (–80) ÷ 80 = –1 (c) (–205) ÷ (–205) = 1(d) –89 ÷ (–1) = 89 (e) (–83) ÷ 1 = –83 (f) (–48) ÷ 48 = –1(g) 30 ÷ (–15) = –2 (h) (–12) ÷ (3) = –4

Example 27: Write five pairs of integers (a, b) such that b ÷ a = –2. One such pair is (3, –6) because–6 ÷ 3 = –2.

Solution: The required pairs of integers are (1, –2), (–2, 4), (4, –8), (–5, 10) and (7, –14).Note: There are infinite pairs of integers (a, b) satisfying b = –2a.

Example 28: The temperature at noon was 10°C. If it decreases at the rate of 2°C per hour untilmidnight, at what time would the temperature be 6°C below zero? What would be thetemperature at midnight?

Solution: Temperature at noon = +10°CRequired temperature = –6°C

14

One way to find values of C and I is by hitand trial method. We assume some values ofC and I so that their sum is 16 (total questionsattempted).It is clear from the table that Rakesh attempted10 questions correctly and 6 questionsincorrectly.

Example 30: An elevator descends into a mine shaft at the rate of 8 m per min. If the descent startsfrom 16 m above the ground level, how long will it take to reach –240 m?

Solution: Total distance to be covered = 16 – (–240) = 256 mElevator descends at the rate of 8 m per min

∴ Required time to reach at –240 m = (240 16) m

8 m/min

+ = 32 min

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1. Divide the following:(a) 75 by –5 (b) –36 by –12 (c) 34 by –17 (d) 0 by –98

2. Fill in the blanks:(a) 45 ÷ ______ = –5 (b) ______ ÷ (–12) = 5 (c) 23 ÷ ______ = –1(d) –11 ÷ ______ = 1 (e) ______ ÷ (–2) = 38

3. State true or false for each of the following:(a) 0 ÷ 7 = 0 (b) (–18) ÷ 0 = 0 (c) 0 ÷ 0 = 0 (d) (–10) ÷ (–2) = 5

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So far we have learnt the basic fundamental operations of addition, subtraction, multiplication anddivision. Mathematical expressions having only one operation can be simplified starting from leftto right as shown: 5 + 7 + 6 + 10 = 28 or 2 × 3 × 4 × 5 = 120

When we have many operations to be performed then simplifying as we did in the above examplescan be quite confusing and may give wrong results. For example, does 24 ÷ 6 × 2 mean (24 ÷ 6) ×2 = 4 × 2 = 8 or 24 ÷ (6 × 2) = 24 ÷ 12 = 2? In such a situation we need to use B O D M A S rule.

B ⎯⎯⎯⎯⎯→ BRACKETS

O ⎯⎯⎯⎯⎯→ OF

D ⎯⎯⎯⎯⎯→ DIVISION

M ⎯⎯⎯⎯⎯→ MULTIPLICATION

A ⎯⎯⎯⎯⎯→ ADDITION

S ⎯⎯⎯⎯⎯→ SUBTRACTION

BODMAS denotes the order of operations to be performed. According to it, first the brackets aresolved, then ‘of’ followed by the four operations of division, multiplication, addition and subtraction.The operations cannot be performed in the order in which they appear in an expression.

C I Marks scored3(C) – 2(I)

8 8 3(8) – 2(8) = 89 7 3(9) – 2(7) = 13

10 6 3(10) – 2(6) = 18

(+, ×, ÷, –)or

(÷, ×, +, –)

[{(...) + 2} + 2]

[()]

15

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Let’s learn about different type of brackets used in Arithmetic. The brackets are removed in theorder mentioned below:

Symbol Name____ Vinculum( ) Round brackets or parenthesis{ } Curly brackets or braces[ ] Square or box brackets

To simplify expressions having brackets the following order is used:(a) If a vinculum is present, then first perform operations under it.(b) If there are brackets within brackets, remove the innermost brackets first by doing all calculations

inside it. Then solve within the next innermost bracket and so on.

Important rules for solving the brackets

• If there is a ‘+’ sign outside the bracket remove it keeping the signs of the terms inside thebracket as it is.

• If there is a ‘–’ sign outside the bracket remove it and change the signs of all the terms insidethe bracket.

• If there is a number just outside the bracket with no sign between the number and thebracket, we multiply the number with all the terms inside the bracket as it meansmultiplication.

Example 31: Simplify 36 – [5 + {27 – (16 – 9)}]Solution: Using BODMAS rule and order of brackets,

36 – [5 + {27 – (16 – 9)}] = 36 – [5 + {27 – 7}] (Removing parenthesis)= 36 – [5 + 20] (Removing curly brackets)= 36 – 25 = 11 (Removing square brackets)

Example 32: Simplify 49 ÷ [49 + {49 – (49 + 49 – 49) }]

Solution: Using BODMAS rule and order of brackets,

49 ÷ [49 + {49 – (49 + 49 – 49) }] = 49 ÷ [49 + {49 – (49 + 0)}]

(Removing vinculum)= 49 ÷ [49 + {49 – 49}] (Removing parenthesis)= 49 ÷ [49 + 0] (Removing curly brackets)= 49 ÷ 49 = 1 (Removing square brackets)

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1. Simplify each of the following:(a) –25 + 12 ÷ (9 – 3) (b) 29 – [38 – {40 ÷ 2 – (6 – 9 ÷ 3) ÷ 3}]

(c) 14 – 1

2{13 + 2 – (7 + 5 – 2 3+ )} (d) 14 +

1

5[{–10 × (25 – 13 3− )} ÷ (–5)]

(e) 27 – 1

4{–5 – (–48) ÷ (–16)}

16

2. Using appropriate brackets, write a mathematical expression for each of the following:(a) Sixty-three divided by one more than the sum of three and five(b) Three multiplied by two less than the difference of sixteen and four(c) Eighteen subtracted from one third the sum of sixty and thirty(d) Eighty divided by four times the sum of two and three(e) Nine multiplied by the sum of three, four and five

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1. The whole numbers along with their negatives, i.e., …, –4, –3, –2, –1, 0, 1 , 2 , 3, … constituteintegers.

2. Absolute value of an integer is always positive. It is the distance of any integer from 0.3. Addition: For any integers a and b, we define, –a + (–b) = –(a + b)

Properties(a) Closure: a + b is again an integer. (b) Commutative: a + b = b + a(c) Associative: (a + b) + c = a + (b + c)(d) Existence of additive identity: Since a + 0 = a = 0 + a for every integer a, ∴ 0 is the

additive identity.(e) Existence of additive inverse: If a + (–a) = 0 = (–a) + a, then –a is called the additive

inverse of a.4. Subtraction: If a and b are any integers, we define a – b = n iff a = b + n, also a – b = a + (–b)

Properties(a) Closure: a – b is again an integer.Commutative and associative properties are not true for subtraction in integers.

5. Multiplication: For any integers a and b, we define (–a)(–b) = ab and (–a)(b) = (a)(–b) = –(ab)Properties(a) Closure: a × b is again an integer. (b) Commutative: a × b = b × a(c) Associative: (a × b) × c = a × (b × c) (d) Distributive: a × (b + c) = (a × b) + (a × c)(e) Existence of multiplicative identity: Since a × 1 = a = 1 × a for every integer a, ∴ 1 is the

multiplicative identity.6. Division: If a and b ≠ 0 are any integers, then a is divisible by b, i.e., a ÷ b, if there exists a

unique integer c such that a = bc.Note: (a) For any integer a, 0 ÷ a = 0 and a ÷ 0 is not defined.

(b) The properties, i.e., closure, commutative and associative are not true for integers.7. Order of operations: When division, multiplication, addition and subtraction appear in an

expression without brackets, multiplication and division are done first in order of their appearancefrom left to right followed by addition and subtraction in order of their appearance from left toright. Any calculation in parenthesis is done first or one can follow the order given in BODMAS.

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Given below are the mathematical barrels which are capable of performing the fundamentaloperations marked on them. When a number is put from the top as input, the operation marked on

17

the label is performed and the answer comes out of the outlet as output. Series (a) is solved as anillustration. Complete series (b) and (c).

(a)

(b)

(c)

(i) Can you find a single operation which transforms the initial input to the final output? Forexample, in part (a) 18 – 17 = 1. Can it be done differently?

(ii) Why does solving expression of the input followed by all the marked operations givedifferent answer? For example, in part (a) 18 + 12 ÷ (–3) × 5 + 53 – 2 = 49 which is differentfrom the final output 1 .

[Hint: The mathematical barrels do not follow the order of BODMAS.]

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1. If the integers 12, –6, 8, –5, 7, –4 and 3 are marked on the number line, the one that comes onthe extreme left is:(a) 12 (b) –4 (c) –6 (d) 3

2. Addition of integers satisfies the property of:(a) closure (b) commutativity (c) associativity (d) all of these

3. Multiplicative identity for any integer a is:

(a) 0 (b) 1 (c) a (d)1

a4. If an expression contains integers connected with the four fundamental operations, then the

correct order in which the operations are to be performed is:(a) ÷, +, –, × (b) ×, ÷, +, – (c) –, +, ×, ÷ (d) ÷, ×, +, –

+12 ÷(–3) ×5 +53 –2

×1 +2 –2 ×(–2) ÷(–2)

–18 ×3 +25 ÷(–5) +2

18

2

13

30 –10 –50 3 1

18

5. The value of the expression 10,000 ÷ {(80 + 100 ÷ 5) × 100} is:(a) 0 (b) 1 (c) 10 (d) 100

6. The difference in temperatures +50°C and –50°C is:(a) 100°C (b) 0°C (c) 50°C (d) –100°C

7. Let a, d and k be integers such that d divides a, then d also divides:(a) a + k (b) a – k (c) ak (d) none of these

8. If 5 divides integer p and 5 does not divide integer r, then:(a) 5 divides (p + r) (b) 5 divides (p – r)(c) 5 does not divide (p + r) (e) 5 does not divied (pr)

9. If an expression involves only addition and subtraction, we cannot:(a) work them from left to right as it comes(b) group the positive and negative integers separately(c) do all the additions first followed by subtraction(d) ignore all the signs and add

10. Additive inverse of (pqrs) where p, q, r and s are non-zero integers is:(a) (–p)(–q)(–r)(–s) (b) (–p)qr(–s) (c) pqrs (d) (–p)(–q)(–r)s

11. The integer x for which |1 – x| = 3 is:(a) 3, 0 (b) 4, 2 (c) –2, 3 (d) 4, –2

12. The number of integers between –30 and –15 are:(a) 14 (b) 15 (c) 16 (d) 17

13. The sum of five consecutive positive integers is always divisible by:(a) 2 (b) 3 (c) 5 (d) 10

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True or False

1. The sum of two integers is always greater than their difference.2. On adding an integer to its additive inverse, we get additive identity.3. For any two integers a and b the inequality –a < b is always true.4. If we divide an integer by –1 the result is negative of the integer.

5. If a is any non-zero integer, then 0

a is not defined.

6. If a is a non-zero integer, then 0

a is not defined.

7. The value of the expression (–12) × (–13) × (–15) × (–3) is smaller than (–13) × 14 × (–16) ×17 × (–18).

8. Associative property of multiplication gives due importance to the order of multiplication ofintegers.

9. Associative property is valid for division of integers.10. When two negative integers are divided the sign of the resulting quotient is positive.11. Division of integers satisfies closure property.12. If a, b, c are integers, then a ÷ (b + c) = (a ÷ b) + (a ÷ c).13. If a, b, c are integers, then (a + b) ÷ c = (a ÷ c) + (b ÷ c).

19

14. If multiplication and division are the only operations in an expression, we can calculate fromleft to right irrespective of the order of operations.

15. For every integer x, |x| is either positive or zero.16. If a and b are integers, then (a – b)2 = (b – a)2.

Fill in the Blanks

1. Sign of the product of two integers that are opposite in sign is ___________.2. Additive inverse of an integer a is ______.3. The value of the expression (–3) × (–1) × (–2) is ______.4. The value of the expression (–45) ÷ 15 ÷ (–3) is ______.5. There are __________ pairs of integers satisfying a ÷ b = –2.6. The distance of any integer from zero is equal to its _____________.

Use the correct symbol >, < or = in question numbers 7 to 9.

7. (–10) + (50) 30 + (–100)

8. (–18) × 0 × 7 (–12) × (–5)

9. (–10) × (–9) × (–8) (–200) + (–240) + (–280)

In question numbers 10 and 11 write an integer in the box to make them true.

10. × (4 + 9) = 7 × 4 + 7 × 9

11. (–7) × 4 + (–7) × = –7 × (4 + 2)

12. Use the parenthesis ( ) only once around two of the integers and put each of the symbols from+, × and ÷ in the boxes to make the following equation true.

8 2 1 6 = 4

Answer Orally

1. Which two integers have modulus value 3?2. Name the identity element of integers with respect to addition.3. If even number of negative integers are multiplied, what will be the sign of the product?4. If odd number of negative integers are multiplied, what will be the sign of the product?5. Which integer when multiplied by –1 gives 23?6. Find the integer which when divided by –1 gives –13.7. What will be the sign of product of 3 negative and 3 positive integers?8. What is the product of any integer with 0?9. What are the order of operations represented by BODMAS?

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1. Write a negative integer and a positive integer whose difference is 13.2. Write a pair of negative integers whose difference is 16.3. Find the products in each case:

(a) (–16) × (–4) × (–3) (b) (–10) × (4) × 0 × (–36) (c) (–12) × (–5) × (–2) × (–10)

20

4. Simplify using suitable properties:(a) 12 × (–3) + 5 × (–3) (b) (–10) × (–7) + (–10) × (–3) (c) 11 × (–68) + 11 × (–32)

5. Morning temperature on a particular day of October in Kashmir was recorded as 3.5ºC.However, by afternoon it rose by 5ºC. What was the temperature recorded in the afternoon?

6. In a quiz, +5 marks are awarded for each correct answer and –5 for each incorrect answer. Infour rounds of the quiz, Surbhi scored 55, –35, 40 and –20 marks and Reetika scored–40, –10, 25 and 35 marks. Who scored higher marks?

7. A cement company earns a profit of � 18 per bag of white cement sold and a loss of � 5 per bagof grey cement sold.(a) The company sells 2,000 bags of white cement and 3,000 bags of grey cement in a month.

What is its profit or loss?(b) If the company sells 3,600 grey bags, find the number of white cement bags it must sell to

make no profit no loss.8. An elevator descends into a mine shaft at the rate of 9 m per min. If the descent starts from

20 m above the ground level, how long will it take to reach –250 m?9. Simplify:

(a) 19 + 1

5[{–20 × (55 –13 – 3) } ÷ (–5)] (b) 38 – 2 (5 – 8 – 3) ÷ [2{7 + (–3) × (–4)}]

10. The temperature of a piece of metal is –13°C. On heating, the temperature of the metal risesto 63°C:(a) What is the change in temperature of the metal piece?(b) What temperature of the metal would be exactly midway between the two temperatures

(the initial and final)?(c) Calculate the temperature of the metal if it rises by 63°C.

11. A chair costs � 120. Peter bought 30 chairs. By mistake the shopkeeper charged � 10 less perchair. To be fair to all, find whether Peter had to pay or receive the money. What is the extraamount?

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1. The share price of a company dropped by 50 points but recovered 40 points the next day. Whatis the net change in its price?

2. Gaurav’s bank balance was � 2,000. In a month he deposited � 300, � 750 and � 450 and issuedcheques worth � 700 and � 1,250. Find the balance in Gaurav’s account at the end of the month?

3. Imagine that you are a fighter pilot flying a jet aircraft at an altitude of 20,000 feet. You lowerthe aircraft by 7,000 feet and then raise it by 5,000 feet. What is the net change in the altitude?Also find the altitude after the two changes.

4. Raghuraj has 30 marbles. In 5 games he won 3 marbles each and in 7 games he lost 2 marbleseach. Find the marbles Raghuraj has at the end of the games.

5. To convert temperature from Celsius to Fahrenheit, we multiply by 9

5 and add 32, i.e.,

F = 9C

325

+ . Find the Fahrenheit temperature that corresponds to:

(a) –15°C (b) 40°C (c) –40°C

21

6. Give an example to show that the following properties are not true for integers.(a) Closure property for division (b) Associative property for division(c) Commutative property for subtraction (d) Distributive property of division over addition

7. Complete the following tables by performing the required operation.

(a) Multiplication: Multiplythe integers in thehorizontal top row withthose in the vertical leftcolumn.

× 1 –2 3 –4

0 0

1 –2

3 9

÷ 0 6 –12 18

1 0

–2 6

–6 –1

(b) Division: Dividethe integers in thehorizontal top rowby the integersgiven in the verticalleft column.

8. In a Microbiology laboratory, certain tests are done under controlled temperature. The variationin temperature is by fixed number of degrees per minute. Evaluate each of the following:(a) The temperature at the beginning of an experiment is 20°C. It increases by 4°C per minute.

What is the temperature at the end of the experiment if it took 15 minutes to complete?(b) The temperature is made to fall by 3°C per minute. The temperature at the end of an

experiment is –7°C, which took 13 minutes to complete. What was the temperature at thebeginning of the experiment?

(c) The temperature at 3 p.m. is 0°C. What can you say about the temperature:(i) m minutes before, if it increases by 5°C per minute.

(ii) 10 minutes later, if it decreases by x°C per minute.9. A pharmaceutical company prepares different types of injections and medicinal creams which

are stored at a fixed range of temperatures.The names of these injections and medicinal creams are coded as INJ-T20 or CRM-5X2, etc.The following graph shows the temperatures, in degrees Celsius at which these are to be stored.Using the information given below, find the injections and medicinal creams that can be storedin a place where the variation in temperature is as follows?(a) Between 0° and 12° (b) Between –6° and 12° (c) Between 9° and 21°(d) Between –9° and 30° (e) Above 15°

–9° –6° –3° 0° 3° 6° 9° 12° 15° 18° 21° 24° 27° 30°

INJ-T20

INJ-F308

INJ-58R

CRM-5X2

CRM-77D

Temperature in °C