## 1.4 DIVISION OF INTEGERS:

**1.4 DIVISION OF INTEGERS:**

If a and b are two integers such that b ≠ 0 **then a ÷ b may or may not **be an integer.

If a is an integer and a ≠ 0 then **a ÷ −a = −1. **

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

If a is an integer and a ≠ 0 then **0 ÷ a = 0 .**

**EXERCISE 1.4**

**Evaluate each of the following:**

a) (–30) ÷ 10 b) 50 ÷ (–5)

c) (–36) ÷ (–9) d) (–49) ÷ (49)

e) 13 ÷ [(–2) + 1] f) 0 ÷ (–12)

g) (–31) ÷ [(–30) + (–1)] h) [(–36) ÷ 12] ÷ 3

i) [(–6) + 5] ÷ [(–12) + 1]

**2) Verify that a÷ (b + c) ≠ (a ÷ b) + (a ÷ c)**** for each of the following values of a, b and c.**

a) a = 12, b= –4, c = 2

b) a = –10, b = 1, c= 1

**Solution:** Given that a = 12, b = –4 and c = 2

L.H.S = a ÷ (b + c) = 12 ÷ (–4 + 2)

= 12 ÷ (–2) = –6

R.H.S = (a ÷ b) + (a ÷ c) = [12 ÷ (–4)] + (12 ÷ 2)

= (–3) + 6 = 3

∴ –6 ≠ 3

∴ ** L.H.S ≠ R.H.S**

Hence, a ÷ (b ÷ c) ≠ (a ÷ b) + (a ÷ c)

b) Given that a = –10, b = 1, c= 1

L.H.S = a ÷ (b + c) = (–10) ÷ (1 + 1)

= (–10) ÷ 2 = –5

R.H.S = (a ÷ b) + (a ÷ c) = [(–10) ÷ 1] + [(–10) ÷ 1]

= (–10) + (–10) = –20

∴ –5 ≠ –20

∴ **L.H.S ≠ R.H.S**

Hence, a ÷ (b ÷ c) ≠ (a ÷ b) + (a ÷ c)

**3) Fill in the blanks: **

a) 369 ÷ ____ = 369 b) (−75) ÷ ___ = −1

c) (−206) ÷ ____ = 1 d) −87 ÷ _____ = 87

e) ____ ÷ = −87 f) ____÷ 48 = −1

g) 20 ÷ ____ = −2 h) ____ ÷ 4 = −3

**Solution:**

a) ∴ a ÷ 1 = a ∴ 369 ÷ 1 =** 369**

b) ∴ (−a) ÷ a = −1 ∴ (−75) ÷ 75 =** −1**

c) ∴ (−a) ÷ −a = 1 ∴ (−206) ÷ (−206) **= 1**

d) ∴ (−a) ÷ (−1) = a ∴ (−87) ÷ (−1) **= 87**

e) ∴ (−a) ÷ a = −a ∴ (−87) ÷ 1 = **− 87**

f) ∴ (−a) ÷ a = −1 ∴ (−48) ÷ 48 **= −1**

g) ∴ 20 ÷ 10 = 2 ∴ 20 ÷ (−10) =** −2**

h) ∴ 12 ÷ 4 = 3 ∴ (−12) ÷ 4 =** −3**

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

**Solution: **

** i) Since, 3 ÷ 1 = 3**

∴ 3 ÷ (–1) = –3

Comparing it with a ÷ b = –3 we have a = 3 and b = (–1)

∴ The required pair of integers **= (3, –1)**

**ii) Since, 3 ÷ 1 = 3**

∴ (–3) ÷ 1 = –3

Comparing it with a ÷ b = –3 we have a = 9 and b = 1

∴ The required pair of integers **= (–3, 1)**

** iii) Since, 9 ÷ 3 = 3**

∴ 9 ÷ (–3) = –3

Comparing it with a ÷ b = –3 we have a =9 and b = –3

∴ The required pair of integers =** (9, –3)**

**iv) Since, 12 ÷ 4 = 3**

∴ (–12) ÷ 4 = –3

Comparing it with a ÷ b = –3 we have a = –12 and b = 4

∴ The required pair of integers =** [–12, 4]**

**v) Since, 12 ÷ 4 = 3**

∴ 12 ÷ (–4) = –3

Comparing it with a ÷ b = –3 we have a = 12 and b = –4

∴ The required pair of integers =** [12, (–4)]**

**5. The temperature at 12 noon was 10 ^{o}C above zero. If it decreases at the rate of 2^{o}C per hour until midnight, at what time would the temperature be 8°C below zero? What would be the temperature at mid-night?**

**Solution:-**

Temperature at 12 noon = + 10०C

Rate of change in temperature = – 2^{o}C per hour

Number of hours from 12 noon to mid – night = 12

∴ Change in temperature in 12 hours = 12० x (–2) = –24०C

∴ Temperature at mid-night (i.e. 12 hours after 12 noon) = +10०C + (24०C)

= – 14^{o}C

Thus, temperature at mid- night = – 14०C

Now temperature difference between + 10०C and – 8 = + 10०C – (–8०C)

= 18०C

∴ Temperature change of 18०C will take place in 9 hours from 12 noon.

Time after 9 hours from 12 noon = 9 p.m.

**Thus, the temperature 8०C below 0० (8०C) would be at 9 p.m.**

**6. In a class test (+ 3) marks are given for every correct answer and (–2) marks are given for every incorrect answer and no marks for not attempting any question.**

** (i) Radhika scored 20 marks. If she has got 12 correct answers, how many questions has she attempted incorrectly?**

**(ii) Mohini scores –5 marks in this test, though she has got 7 correct answers. How many questions has she attempted incorrectly?**

**Solution:- **

**Marks for every correct answer = + 3**

**Marks for every incorrect answer = –2**

(i) Total marks scored by Radhika = 20

Marks scored for correct answers = 12 × 3 = 36

∴ Marks given for incorrect answers = 20 – 36 = – 16

∴ Number of incorrect answers = (–16) ÷ (–2) = 8

**Thus, Radhika attempted 8 questions incorrectly.**

(ii) Marks obtained by Mohini = –5

Marks obtained for 7 correct answers = 7 × 3 = 21

∴ Marks obtained for incorrect answers = – 5 – 21 = – 26

∴ Number of incorrect answers = (–26) ÷ (–2) = 13

**Thus, Mohini attempted 13 questions incorrectly.**

**7. An elevator descends into a mine shaft at the rate of 6 m/min. If the descent starts from 10 m above the ground level, how long will it take to reach – 350 m.**

**Solution:-**

Present position of the elevator is at 10 m above the ground level.

Distanced to be moved by the elevator below the ground level = 350 m

∴ Total distance to be moved by the elevator = 350 m + 10 m

= 360 m