Exercise 1.1

Question: 1

Determine each of the following products:

(i) 12 × 7

(ii) (-15) × 8

(iii) (- 25) × (- 9)

(iv) (125) × (- 8)

Solution:

(i) We have,

12 × 7 = 84  [The product of two integers of like signs is equal to the product of their absolute value]

(ii) We have,

 (- 15) × 8 [The product of two integers of opposite

= (- 15 × 8) signs is equal to the additive inverse of the

= –120 [product of their absolute values]

(iii) We have,

(-25) × (-9)

= + (25 × 9)

= 225

(iv) We have,

(125) × (- 8)

= – (125 × 8)

= –1000

Question: 2

Find each of the following products:

(i) 3 × (- 8) × 5

(ii) 9 × (- 3) × (- 6)

(iii) (- 2) × 36 × (- 5)

(iv) (- 2) × (- 4) × (- 6) × (- 8)

Solution:

(i) We have,

3 × (- 8) × 5

= – (3 × 8) × 5

= (- 24) × 5

= – (24 × 5)

= – 120

(ii) We have,

9 × (-3) × (- 6)

= – (9 × 3) × (- 6)

= (- 27) × (- 6)

= + (27 × 6)

= 162

(iii) We have,

(-2) × 36 × (- 5)

= – (2 × 36) × (- 5)

= (- 72) × (- 5)

= (72 × 5)

= 360

(iv) We have,

(- 2) × (- 4) × (- 6) × (- 8)

= (2 × 4) × (6 × 8)

= (8 × 48)

= 384

Question: 3

Find the value of:

(i) 1487 × 327 + (- 487) × 327

(ii) 28945 × 99 – (- 28945)

Solution:

(i) We have,

1487 × 327 + (- 487) × 327

= 486249 – 159249

= 327000

(ii) We have,

28945 × 99 – (- 28945)

= 2865555 – 28945

= 2894500

Question: 4

Complete the following multiplication table:

Second Number

X	– 4	-3	– 2	-1	0	1	2	3	4
– 4
First Number
– 2
-1
0
1
2
3
4

Is the multiplication table symmetrical about the diagonal joining the upper left corner to the lower right corner?

Solution:

Second number

X	– 4	– 3	– 2	– 1	0	1	2	3	4
– 4	16	12	8	4	0	-4	-8	-12	-16
First number	12	9	6	3	0	-3	-6	-9	-12
-3
-2	8	6	4	2	0	-2	-4	-6	-8
-1	4	3	2	1	0	-1	-2	-3	-4
0	0	0	0	0	0	0	0	0	0
1	-4	-3	-2	-1	0	1	2	3	4
2	-8	-6	-4	-2	0	2	4	6	8
3	-12	-9	-6	-3	0	3	6	9	12
4	-16	-12	-8	-4	0	4	8	12	16

Question: 5

Determine the integer whose product with ‘-1’ is

(i) 58

(ii) 0

(iii) – 225

Solution:

(i) 58 x (–1) = – (58 x 1)

= – 58

(ii) 0 x (–1) = 0

(iii) (–225) x (–1) = + (225 x 1)

= 225

Question: 6

What will be the sign of the product if we multiply together

(i) 8 negative integers and 1 positive integer?

(ii) 21 negative integers and 3 positive integers?

(iii) 199 negative integers and 10 positive integers? 

Solution:

(i) Positive ∵ [- ve × – ve = + ve]

(ii) Negative ∵ [- ve × + ve = – ve]

(iii) Negative

Question: 7

State which is greater: 

(i) (8 + 9) × 10 and 8 + 9 × 10

(ii) (8 – 9) × 10 and 8 – 9 × 10 

(iii) ((-2) – 5) × – 6 and (-2) – 5 × (- 6)

Solution:

(i) (8 + 9) × 10 = 17 × 10

= 170

8 + 9 × 10 = 8 + 90 = 98

(8 + 9) × 10 > 8 + 9 × 10

(ii) (8 – 9) × 10 = – 1 × 10

= – 10

8 – 9 × 10 = 8 – 90 = – 82

(8 – 9) × 10 > 8 – 9 × 10

(iii)  ((-2) – 5) × – 6 = (- 7) × (- 6)

= (7 x 6)

= 42

(– 2) – 5 x (– 6) = – 2 + (5 x 6)

= 30 – 2

= 28

Therefore, ((-2) – 5×(- 6)) > (- 2) – 5 × (- 6)

Question: 8

(i) If a× (-1) = – 30, is the integer a positive or negative?

(ii) If a × (-1) = 30, is the integer a positive or negative? 

Solution:

(i) When multiplied by ‘a’ negative integer, a gives a negative integer. Hence, ‘a’ should be

a positive integer.

a = 30

(ii) When multiplied by ‘a’ negative integer, a gives a positive integer. Hence, ‘a’ should be

a negative integer.

a = – 30

Question: 9

 Verify the following:

(i) 19 × (7 + (-3)) = 19 × 7 + 19 × (-3)

(ii) (-23)[(-5)+ (+19)] = (-23) × (- 5) + (- 23) × (+19)

Solution:

(i) L.H.S = 19 × (7+ (-3))

= 19 × (7-3)

= 19 × 4

= 76

R.H.S = 19 × 7 + 19 × (-3)

= 133 – 57

= 76

Therefore, L.H.S = R.H.S

(ii)  L.H.S = (-23)[(-5) + (+19)]

= (-23)[-5 + 19]

= (-23)[14]

= – 322

R.H.S = (-23) × (-5) + (-23) × (+19)

= 115 – 437

= –322

Therefore, L.H.S = R.H.S

Question: 10

Which of the following statements are true?

(i) The product of a positive and a negative integer is negative.

(ii) The product of three negative integers is a negative integer.

(iii) Of the two integers, if one is negative, then their product must be positive.

(iv) For all non-zero integers a and b, a × b is always greater than

either a or b.

(v) The product of a negative and a positive integer may be zero.

(vi) There does not exist an integer b such that for a >1, a × b = b × a = b. <

Solution:

(i) True

(ii) True

(iii) False

(iv) False

(v) False

(vi) True

Exercise 1.2

Question: 1

Divide:

(i) 102 by 17

(ii) –85 by 5

(iii) –161 by –23

(iv) 76 by –19

(v) 17654 by –17654

(vi) (–729) by (–27)

(vii) 21590 by – 10

(viii) 0 by –135

Solution:

(i) We have,

(ii) We have,

(iii) We have,

(iv) We have,

(v) We have,

(vi) We have,

(vii) We have,

(viii) We have,

Question: 2

 Fill in the blanks:

 (i) 296 ÷ …. = – 148

(ii) – 88 ÷ …. = 11

(iii) 84 ÷ …. = 12

(v) …. ÷ 156 = – 2

(vi)  …. ÷ 567 = – 1

Solution:

(i)  We have,

(ii) We have,

(iii) 84/12 = 7

(iv)  …. ÷ -5 = 25

25 × (- 5) = – 125

(v) …. ÷ 156 = – 2

– (156 × 2)

= – 312

(vi) x/567 = – 1

⇒ x = – (567 × 1)

= – 567

∴ – (567/567) = -1

Question: 3

Which of the following statements are true?

(i) 0 ÷ 4 = 0

(ii) 0 ÷ (-7) = 0

(iii) -15 ÷ 0 = 0

(iv) 0 ÷ 0 = 0

(v) (- 8) ÷ (- 1) = – 8

(vi) – 8 ÷ (- 2) = 4<

Solution:

(i) True

(ii) True

(iii) False

(iv) False

(v) False

(vi) True

Exercise 1.3

Question: 1

Find the value of 36 ÷ 6 + 3.

Solution:

36 ÷ 6 + 3 = 6 + 3

= 9

Question: 2

Find the value of 24 + 15 ÷ 3.

Solution:

24 + 15 ÷ 3 = 24 + 5

= 29

Question: 3

Find the value of 120 – 20 ÷ 4.

Solution:

120 – 20 ÷ 4 = 120 – 5

= 115

Question: 4

Find the value of 32 – (3 × 5) + 4.

Solution:

32 – (3 x 5) + 4 = 32 – 15 + 4

= 17 + 4

= 21

Question: 5

Find the value of  3 – (5 – 6 ÷ 3).

Solution:

3 – (5 – 6 ÷ 3) = 3 – (5 – 2)

= 3 – 3

= 0

Question: 6

Find the value of 21 – 12 ÷ 3 × 2.

Solution:

21 – 12 ÷ 3 × 2 = 21 – 123 × 2

= 21 – 4 × 2

= 21 – 8

= 13

Question: 7

Find the value of 16 + 8 ÷ 4 – 2 × 3.

Solution:

16 + 8 ÷ 4 – 2 × 3

= 16 + 2 – 6

= 18 – 6

= 12

∴ 16 + 8 ÷ 4 – 2 × 3 = 12

Question: 8

Find the value of 28 – 5 × 6 + 2.

Solution:

28 – 5 × 6 + 2 = 28 – (5 × 6) + 2

= 28 – 30 + 2

= 30 – 30

= 0

Question: 9

Find the value of  (-20) × (-1) + (-28) ÷ 7.

Solution:

(-20) × (-1) + (-28) ÷ 7 = 20 + ∣-28∣ ∣7∣

= 20 – 287

= 20 – 4

= 16

Question: 10

Find the value of  (-2) + (-8) ÷ (- 4).

Solution:

(-2) + (-8) ÷ (- 4) = – 2 + ∣- 8∣ ∣- 4∣

= – 2 + 2

= 0

Question: 11

Find the value of (- 15) + 4 ÷ (5 – 3).

Solution:

-15 + 4 ÷ (5 – 3) = – 15 + 4 ÷ 2

= – 15 + 2

= – 13

-15 + 4 ÷ (5 – 3) = – 13

Question: 12

Find the value of (- 40) × (-1) + (-28) ÷ 7.

Solution:

(- 40) × (-1) + (-28) ÷ 7 = 40 + (- 4)

= 40 – 4

= 36

Question: 13

Find the value of (-3) + (-8) ÷ (-4) – 2× (-2).

Solution:

(-3) + (-8) ÷ (-4) – 2 × (-2) = (-3) + (-8)(-4) – 2×(-2)

= – 3 + 2 + 4

= 6 – 3

= 3

Question: 14

Find the value of (-3) × (-4) ÷ (-2) + (-1).

Solution:

(-3) × (-4) ÷ (-2) + (-1) = 12 ÷ (-2) + (-1)

= – 6 – 1

= – 7

∴ (-3) × (-4) ÷ (-2) + (-1) = – 7

Exercise 1.4

Question: 1

Simplify

3 – (5 – 6 ÷ 3)

Solution:

3 – (5 – 6 ÷ 3)

= 3 – (5 – 2)

= 3 – 3

= 0

∴ 3 – (5 – 6 ÷ 3) = 0

Question: 2

Simplify

– 25 + 14 ÷ (5 – 3)

Solution:

-25 + 14 ÷ (5 – 3) = – 25 + 14 ÷ (2)

= – 25 + 14/2

= –25 + 7

= –18

∴ – 25 + 14 ÷ (5 – 3) = – 18

Question: 3

Simplify

Solution:

Question: 4

Simplify

Solution:

Question: 5

Simplify

36 – [18 – (14 – (15 – 4 ÷ 2 × 2))]

Solution:

36 – [18 – (14 – (15 – 4 ÷ 2 × 2))]

= 36 – [18 – (14 – (11 ÷ 2 × 2))]

= 36 – [18 – (14 – 11/2 × 2))]

= 36 – [18 – (14 – 11)]

= 36 – [18 – 3]

= 36 – 15

= 21

∴ 36 – [18 – (14 – (15 – 4 ÷ 2 × 2))] = 21

Question: 6

Simplify

45 – [38 – (60 ÷ 3 – (6 – 9 ÷ 3) ÷ 3)]

Solution:

45 – [38 – (60 ÷ 3 – (6 – 9 ÷ 3) ÷ 3)]

= 45 – [38 – (20 – (6 – 3) ÷ 3)]

= 45 – [38 – (20 – 3 ÷ 3)]

= 45 – [38 – (20 – 1)]

= 45 – [38 – 19]

= 45 – [19]

= 26

∴ 45 – [38 – (60 ÷ 3 – (6 – 9 ÷ 3) ÷ 3)] = 26

Question: 7

Simplify

Solution:

= 23 – [23 – (23 – (23 – 0))]

= 23 – [23 – (23 – 23)]

= 23 – [23 – 0]

= 23 – 23

= 0

Question: 8

Simplify

Solution:

= 2550 – [510 – (270 – (90 – 150))]

= 2550 – [510 – (270 – (–60))]

= 2550 – [510 – 330]

= 2550 – [180]

= 2550 – 180

= 2370

Question: 9

Simplify

Solution:

Question: 10

Simplify

Solution:

Question: 11

Simplify

Solution:

Question: 12

Simplify

[29 – ( – 2)(6 – (7 – 3))] ÷ [3 × (5 + ( – 3) × (-2))]

Solution:

[29 – (-2)(6 – (7 – 3))] ÷ [3 × (5 + (- 3) × (- 2))]

= [29 – (- 2)(6 – 4)] ÷ [3 × (5 + (3 × 2))]

= [29 – (-2)(2)] ÷ [3 × (5 + 6)]

= [29 + 4] ÷ [3 × 11]

= [33] ÷ [33]

= 1

∴ [29 – (-2) (6 -(7- 3))] ÷ [3 × (5 + (- 3) × (-2))] = 1

Question: 13

Using brackets, write a mathematical expression for each of the following:

(i) Nine multiplied by the sum of two and five.

(ii) Twelve divided by the sum of one and three.

(iii) Twenty divided by the difference of seven and two.

(iv) Eight subtracted from the product of two and three.

(v) Forty divided by one more than the sum of nine and ten.

(vi) Two multiplied by one less than the difference of nineteen and six.

Solution:

(i) 9 (2 + 5)

(ii) 12 ÷ (1 + 3)

(iii) 20 ÷ (7 – 2)

(iv) 2 × 3 – 8

(v) 40 ÷ [1 + (9 + 10)]

(vi) 2 × [(19 – 6) – 1]