NCERT Class 7 Mathematics Textbook for blind and VI students made Screen Readable by Professor T K Bansal.

Observe the following table and complete it:

What do you observe?
We observe that integers are not closed under division.

Justify it by taking five more examples of your own.

We know that division is not commutative for whole numbers. Let us check it for integers also.

You can see from the table that (− 8) ÷ (− 4) ≠ (−4) ÷ (−8).
Is (− 9) ÷ 3 the same as 3 ÷ (− 9)?
Is (− 30) ÷ (− 6) the same as (− 6) ÷ (− 30)?
Can we say that division is commutative for integers?
No.
You can verify it by taking five more pairs of integers.

Like whole numbers, any integer divided by zero is meaningless and zero divided by an integer other than zero is equal to zero i.e., for any integer a, a ÷ 0 is not defined but 0 ÷ a = 0 for a ≠ 0.

When we divide a whole number by 1 it gives the same whole number. Let us check whether it is true for negative integers also.

Observe the following :
( −8) ÷ 1 = ( −8)
( −11) ÷ 1 = −11
( −13) ÷ 1 = −13
( −25) ÷ 1 = ___
( −37) ÷ 1 = ___
( −48) ÷ 1 = ___

This shows that negative integer divided by 1 gives the same negative integer.

So, any integer divided by 1 gives the same integer.

In general, for any integer a,
a ÷ 1 = a

What happens when we divide any integer by ( −1)?
Complete the following table
(− 8) ÷ (−1) = 8
11 ÷ (−1) = −11
13 ÷ (−1) = ____
(−25) ÷ (−1) = ____
(−37) ÷ (−1) = ____
− 48 ÷ (−1) = ____

What do you observe?

We can say that if any integer is divided by (−1) it does not give the same integer.

Can we say [(−16) ÷ 4] ÷ (−2) is the same as (−16) ÷ [4 ÷ (−2)]?

We know that [(−16) ÷ 4] ÷ (−2) = (− 4) ÷ (−2) = 2
and (−16) ÷ [4 ÷ (−2)] = (−16) ÷ (−2) = 8

So [(−16) ÷ 4] ÷ (−2) ≠ (−16) ÷ [4 ÷ (−2)]

Can you say that division is associative for integers? No.

Verify it by taking five more examples of your own.

TRY THESE 1.14

Is
(i) 1 ÷ a = 1?
(ii) a ÷ (−1) = − a? for any integer a.
Take different values of a and check.

EXAMPLE 6

In a test (+5) marks are given for every correct answer and (−2) marks are given for every incorrect answer.
(i) Radhika answered all the questions and scored 30 marks though she got 10 correct answers.
(ii) Jay also answered all the questions and scored ( −12) marks though he got 4 correct answers.
How many incorrect answers had they attempted?

SOLUTION:

(i) Marks given for one correct answer = 5
So, marks given for 10 correct answers = 5 × 10 = 50
Radhika’s score = 30
Marks obtained for incorrect answers = 30 − 50 = − 20
Marks given for one incorrect answer = (−2)
Therefore, number of incorrect answers = ( −20) ÷ ( −2) = 10

(ii) Marks given for 4 correct answers = 5 × 4 = 20
Jay’s score = −12
Marks obtained for incorrect answers = −12 − 20 = − 32
Marks given for one incorrect answer = ( −2)
Therefore number of incorrect answers = ( −32) ÷ ( −2) = 16

EXAMPLE 7

A shopkeeper earns a profit of Rs.1 by selling one pen and incurs a loss of 40 paisa per pencil while selling pencils of her old stock.
(i) In a particular month she incurs a loss of Rs.5. In this period, she sold 45 pens. How many pencils did she sell in this period?
(ii) In the next month she earns neither profit nor loss. If she sold 70 pens, how many pencils did she sell?

SOLUTION:

(i) Profit earned by selling one pen = Rs.1
Profit earned by selling 45 pens = Rs.45, which we denote by + Rs. 45
Total loss given = Rs.5, which we denote by − Rs.5
Profit earned + Loss incurred = Total loss
Therefore, Loss incurred = Total Loss − Profit earned
= Rs. ( −5 − 45) = Rs. ( −50) = −5000 paisa

Loss incurred by selling one pencil = 40 paisa which we write as − 40 paisa
So, number of pencils sold = ( −5000) ÷ ( − 40) = 125

(ii) In the next month there is neither profit nor loss.
So, Profit earned + Loss incurred = 0
i.e., Profit earned = − Loss incurred.
Now, profit earned by selling 70 pens = Rs.70
Hence, loss incurred by selling pencils = Rs.70 which we indicate by − Rs.70 or − 7,000 paisa.
Total number of pencils sold = ( −7000) ÷ ( −40) = 175 pencils.

EXERCISE 1.4

Q1. 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)] ÷ [( −2) + 1]

A1.
(a) −3
(b) −10
(c) 4
(d) −1
(e) −13
(f ) 0
(g) 1
(h) −1
(i) 1

Q2. 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

Q3. Fill in the blanks:
(a) 369 ÷ ___ = 369
(b) ( −75) ÷ ___ = −1
(c) ( −206) ÷ ___ = 1
(d) − 87 ÷ ___ = 87
(e) ___ ÷ 1 = − 87
(f) ___ ÷ 48 = −1
(g) 20 ÷ ___ = −2
(h) ___ ÷ (4) = −3

A3.
(a) 1
(b) 75
(c) − 206
(d) −1
(e) − 87
(f ) − 48
(g) −10
(h) −12

Q4. Write five pairs of integers (a, b) such that a ÷ b = −3. One such pair is (6, −2) because 6 ÷ (−2) = (−3).

A4. (−6, 2), (−12, 4), (12, − 4), (9, −3), (−9, 3) (There could be many such pairs)

Q5. The temperature at 12 noon was 10°C above zero. If it decreases at the rate of 2°C per hour until midnight, at what time would the temperature be 8°C below zero?
What would be the temperature at mid-night?

A5. 9 p.m.; − 14°C

Q6. 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?

A6.
(i) 8
(ii) 13

Q7. 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.

A7. 1 hour