Exercise 1.2 - Solutions

1. Fill in the blanks:

(i) The value of -512 + 715 = _____

Solution: LCM of 12 and 15 is 60.

= -5 × 560 + 7 × 460 = -25 + 2860 = 360 = 120

Answer: 120

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(ii) The value of (-36) × (18-9) is _____

Solution: (-36) × (-189) = (-12) × (-2) = 1

Answer: 1

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(iii) The value of (-1523) ÷ (30-46) is _____

Solution: (-1523) × (-4630) = (-1530) × (-4623) = (-12) × (-2) = 1

Answer: 1

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(iv) The rational number _____ does not have a reciprocal.

Answer: 0

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(v) The multiplicative inverse of -1 is _____

Answer: -1

2. Say True or False:

(i) All rational numbers have an additive inverse.

Answer: True

(ii) The rational numbers that are equal to their additive inverses are 0 and -1.

Answer: False

(Reason: Only 0 is equal to its additive inverse. Additive inverse of -1 is +1.)

(iii) The additive inverse of -11-17 is 1117.

Answer: False

(Reason: -11-17 = 1117. Its additive inverse is -1117.)

(iv) The rational number which is its own reciprocal is -1.

Answer: True

(Note: 1 and -1 are both their own reciprocals.)

(v) The multiplicative inverse exists for all rational numbers.

Answer: False

(Reason: Multiplicative inverse does not exist for 0.)

3. Find the sum:

(i) 75 + 35

Solution: Same denominators.

= 7 + 35 = 105 = 2

Answer: 2

(ii) 75 + 57

Solution: LCM of 5 and 7 is 35.

= 49 + 2535 = 7435 (or 2 435)

Answer: 7435

(iii) 65 + (-1415)

Solution: LCM of 5 and 15 is 15.

= 18 + (-14)15 = 415

Answer: 415

(iv) -4 23 + 7 512

Solution: Convert to improper fractions: -143 + 8912

LCM of 3 and 12 is 12: -56 + 8912 = 3312 = 114 (or 2 34)

Answer: 114

4. Subtract:

Subtract -844 from -1711

Solution: -1711 - (-844)

Simplify -844 to -211:

= -1711 - (-211) = -17 + 211 = -1511

Answer: -1511

5. Evaluate:

(i) 9132 × -113

Solution: Simplify (93) × (-11132) = 3 × (-112) = -312 = -14

Answer: -14

(ii) -727 × 24-35

Solution: = (-7-35) × (2427) = (15) × (89) = 845

Answer: 845

6. Divide:

(i) -215 by -7-10

Solution: -7-10 = 710

= -215 ÷ 710 = -215 × 107 = (-3) × 2 = -6

Answer: -6

(ii) -313 by -3

Solution: = -313 × (1-3) = 113

Answer: 113

(iii) -2 by -615

Solution: = -2 × (15-6) = -2 × (-52) = 5

Answer: 5

7. Find (a + b) ÷ (a - b) if:

(i) a = 12, b = 23

a + b = 12 + 23 = 3 + 46 = 76

a - b = 12 - 23 = 3 - 46 = -16

(a + b) ÷ (a - b) = 76 ÷ (-16) = 76 × (-6) = -7

Answer: -7

(ii) a = -35, b = 215

a + b = -35 + 215 = -9 + 215 = -715

a - b = -35 - 215 = -9 - 215 = -1115

(a + b) ÷ (a - b) = (-715) ÷ (-1115) = (-715) × (-1511) = 711

Answer: 711

8. Simplify and show that it is a rational number between 11 and 12:

Simplify: 12 + (32 - 25) ÷ 310 × 3

Step 1 (Bracket): 32 - 25 = 15 - 410 = 1110

Step 2 (Division): 1110 ÷ 310 = 1110 × 103 = 113

Step 3 (Multiplication): 113 × 3 = 11

Step 4 (Addition): 12 + 11 = 11 12 = 11.5

Since 11 < 11.5 < 12, it lies between 11 and 12.

Answer: 11.5 (lies between 11 and 12)

9. Simplify:

(i) [118 × (-633)] + [13 + (35 ÷ 920)] - [47 × -75]

Part 1: 118 × (-633) = -14

Part 2: 35 ÷ 920 = 35 × 209 = 43 ⇒ 13 + 43 = 53

Part 3: 47 × (-75) = -45

Combine: (-14) + 53 - (-45) = -14 + 53 + 45

LCM of 4, 3, 5 is 60: -15 + 100 + 4860 = 13360

Answer: 13360

(ii) [43 ÷ (8-7)] - [34 × 43] + [43 × (-14)]

Part 1: 43 × (-78) = -76

Part 2: 34 × 43 = 1

Part 3: 43 × (-14) = -13

Combine: (-76) - 1 + (-13) = -7 - 6 - 26 = -156 = -52

Answer: -52

10. Word Problem:

A student had multiplied a number by 43 instead of dividing it by 43 and got 70 more than the correct answer. Find the number.

Solution: Let the number be x.

Correct operation: x ÷ 43 = 34x

Wrong operation performed: x × 43 = 43x

Given difference: 43x - 34x = 70

16 - 912 x = 70 ⇒ 712x = 70

x = 70 × 127 = 10 × 12 = 120

Answer: 120