NCERT solutions

NCERT Solutions for Class 7 Maths Chapter 1 Integers, Solutions to Exercises 1.1, 1.2, and 1.3

Pushpender Yadav

May 12, 2025 — 8 min read

NCERT Solutions

Exercise 1.1

1. Write down a pair of integers whose:

(a) Sum is -7

We need two integers ( a ) and ( b ) such that ( a + b = -7 ).
Try ( a = -4 ), ( b = -3 ): ( -4 + (-3) = -7 ).
Answer: ( (-4, -3) ).

(b) Difference is -10

We need ( a - b = -10 ).
Try ( a = -7 ), ( b = 3 ): ( -7 - 3 = -10 ).
Answer: ( (-7, 3) ).

(c) Sum is 0

We need ( a + b = 0 ), so ( b = -a ).
Try ( a = 5 ), ( b = -5 ): ( 5 + (-5) = 0 ).
Answer: ( (5, -5) ).

2.

(a) Write a pair of negative integers whose difference gives 8.

We need ( a - b = 8 ), where ( a ) and ( b ) are negative.
Try ( a = -2 ), ( b = -10 ): ( -2 - (-10) = -2 + 10 = 8 ).
Answer: ( (-2, -10) ).

(b) Write a negative integer and a positive integer whose sum is -5.

We need ( a + b = -5 ), where ( a ) is negative, ( b ) is positive.
Try ( a = -7 ), ( b = 2 ): ( -7 + 2 = -5 ).
Answer: ( (-7, 2) ).

(c) Write a negative integer and a positive integer whose difference is -3.

We need ( a - b = -3 ), where ( a ) is negative, ( b ) is positive.
Try ( a = -1 ), ( b = 2 ): ( -1 - 2 = -3 ).
Answer: ( (-1, 2) ).

3. In a quiz, team A scored -40, 10, 0 and team B scored 10, 0, -40 in three successive rounds. Which team scored more? Can we say that we can add integers in any order?

Team A’s total score: ( -40 + 10 + 0 = -30 ).
Team B’s total score: ( 10 + 0 + (-40) = -30 ).
Comparison: Both teams scored -30, so neither scored more.
Can we add integers in any order? Yes, addition is commutative and associative for integers (sections 1.1.3 and 1.1.4). For example, ( -40 + 10 + 0 = 10 + (-40) + 0 = -30 ).
Answer: Both teams scored the same (-30). Yes, integers can be added in any order.

4. Fill in the blanks to make the following statements true:

(i) ( (-5) + (-8) = (-8) + (__) )

By commutative property, ( (-5) + (-8) = (-8) + (-5) ).
Answer: ( -5 ).

(ii) ( -53 + __ = -53 )

We need ( -53 + x = -53 ). So, ( x = 0 ) (additive identity, section 1.1.5).
Answer: ( 0 ).

(iii) ( 17 + __ = 0 )

We need ( 17 + x = 0 ). So, ( x = -17 ) (additive inverse).
Answer: ( -17 ).

(iv) ( [13 + (-12)] + (__) = 13 + [(-12) + (-7)] )

Left side: ( [13 + (-12)] = 1 ). So, ( 1 + x = \text{right side} ).
Right side: ( (-12) + (-7) = -19 ), so ( 13 + (-19) = -6 ).
Thus, ( 1 + x = -6 ), so ( x = -7 ).
Answer: ( -7 ).

(v) ( (-4) + [15 + (-3)] = [-4 + 15] + __ )

Left side: ( 15 + (-3) = 12 ), so ( (-4) + 12 = 8 ).
Right side: ( [-4 + 15] = 11 ), so ( 11 + x = 8 ).
Thus, ( x = 8 - 11 = -3 ).
Answer: ( -3 ).

Exercise 1.2

1. Find each of the following products:

(a) ( 3 \times (-1) )

Positive times negative gives negative: ( 3 \times (-1) = -3 ) (section 1.2.1).
Answer: ( -3 ).

(b) ( (-1) \times 225 )

( (-1) \times 225 = -225 ).
Answer: ( -225 ).

(c) ( (-21) \times (-30) )

Negative times negative gives positive: ( (-21) \times (-30) = 21 \times 30 = 630 ) (section 1.2.2).
Answer: ( 630 ).

(d) ( (-316) \times (-1) )

( (-316) \times (-1) = 316 ) (multiplying by -1 gives the additive inverse, section 1.3.4).
Answer: ( 316 ).

(e) ( (-15) \times 0 \times (-18) )

Any integer multiplied by 0 is 0: ( (-15) \times 0 = 0 ), so ( 0 \times (-18) = 0 ) (section 1.3.3).
Answer: ( 0 ).

(f) ( (-12) \times (-11) \times (10) )

First, ( (-12) \times (-11) = 12 \times 11 = 132 ).
Then, ( 132 \times 10 = 1320 ).
Answer: ( 1320 ).

(g) ( 9 \times (-3) \times (-6) )

First, ( 9 \times (-3) = -27 ).
Then, ( (-27) \times (-6) = 27 \times 6 = 162 ).
Answer: ( 162 ).

(h) ( (-18) \times (-5) \times (-4) )

First, ( (-18) \times (-5) = 18 \times 5 = 90 ).
Then, ( 90 \times (-4) = -360 ).
Answer: ( -360 ).

(i) ( (-1) \times (-2) \times (-3) \times 4 )

First, ( (-1) \times (-2) = 2 ).
Then, ( 2 \times (-3) = -6 ).
Then, ( (-6) \times 4 = -24 ).
Answer: ( -24 ).

(j) ( (-3) \times (-6) \times (-2) \times (-1) )

First, ( (-3) \times (-6) = 18 ).
Then, ( 18 \times (-2) = -36 ).
Then, ( (-36) \times (-1) = 36 ).
Answer: ( 36 ).

2. Verify the following:

(a) ( 18 \times [7 + (-3)] = [18 \times 7] + [18 \times (-3)] )

Left side: ( 7 + (-3) = 4 ), so ( 18 \times 4 = 72 ).
Right side: ( 18 \times 7 = 126 ), ( 18 \times (-3) = -54 ), so ( 126 + (-54) = 72 ).
Since ( 72 = 72 ), the distributive property holds (section 1.3.6).
Answer: Verified.

(b) ( (-21) \times [(-4) + (-6)] = [(-21) \times (-4)] + [(-21) \times (-6)] )

Left side: ( (-4) + (-6) = -10 ), so ( (-21) \times (-10) = 21 \times 10 = 210 ).
Right side: ( (-21) \times (-4) = 84 ), ( (-21) \times (-6) = 126 ), so ( 84 + 126 = 210 ).
Since ( 210 = 210 ), the distributive property holds.
Answer: Verified.

3.

(i) For any integer ( a ), what is ( (-1) \times a ) equal to?

From section 1.3.4, ( (-1) \times a = -a ) (multiplying by -1 gives the additive inverse).
Answer: ( -a ).

(ii) Determine the integer whose product with ( (-1) ) is:
(a) -22

Let the integer be ( x ). Then, ( (-1) \times x = -22 ).
So, ( x = 22 ) (since ( (-1) \times 22 = -22 )).
Answer: ( 22 ).

(b) 37

( (-1) \times x = 37 ).
So, ( x = -37 ) (since ( (-1) \times (-37) = 37 )).
Answer: ( -37 ).

(c) 0

( (-1) \times x = 0 ).
So, ( x = 0 ) (since ( (-1) \times 0 = 0 )).
Answer: ( 0 ).

4. Starting from ( (-1) \times 5 ), write various products showing some pattern to show ( (-1) \times (-1) = 1 ).

Observe the pattern:
- ( (-1) \times 5 = -5 )
- ( (-1) \times 4 = -4 = -5 + 1 )
- ( (-1) \times 3 = -3 = -4 + 1 )
- ( (-1) \times 2 = -2 = -3 + 1 )
- ( (-1) \times 1 = -1 = -2 + 1 )
- ( (-1) \times 0 = 0 = -1 + 1 )
- ( (-1) \times (-1) = 1 = 0 + 1 )
The pattern shows the product increases by 1 as the second factor decreases by 1, confirming ( (-1) \times (-1) = 1 ).
Answer: The pattern ( (-1) \times 5 = -5 ), ( (-1) \times 4 = -4 ), ..., ( (-1) \times (-1) = 1 ) shows ( (-1) \times (-1) = 1 ).

Exercise 1.3

1. Evaluate each of the following:

(a) ( (-30) \div 10 )

Negative divided by positive gives negative: ( (-30) \div 10 = -3 ) (section 1.4).
Answer: ( -3 ).

(b) ( 50 \div (-5) )

Positive divided by negative gives negative: ( 50 \div (-5) = -10 ).
Answer: ( -10 ).

(c) ( (-36) \div (-9) )

Negative divided by negative gives positive: ( (-36) \div (-9) = 36 \div 9 = 4 ).
Answer: ( 4 ).

(d) ( (-49) \div 49 )

Negative divided by positive gives negative: ( (-49) \div 49 = -1 ).
Answer: ( -1 ).

(e) ( 13 \div [(-2) + 1] )

First, ( (-2) + 1 = -1 ).
Then, ( 13 \div (-1) = -13 ).
Answer: ( -13 ).

(f) ( 0 \div (-12) )

Zero divided by any non-zero integer is 0: ( 0 \div (-12) = 0 ) (section 1.5).
Answer: ( 0 ).

(g) ( (-31) \div [(-30) + (-1)] )

First, ( (-30) + (-1) = -31 ).
Then, ( (-31) \div (-31) = 1 ).
Answer: ( 1 ).

(h) ( [(-36) \div 12] \div 3 )

First, ( (-36) \div 12 = -3 ).
Then, ( (-3) \div 3 = -1 ).
Answer: ( -1 ).

(i) ( [(-6) + 5] \div [(-2) + 1] )

First, ( (-6) + 5 = -1 ).
Then, ( (-2) + 1 = -1 ).
So, ( (-1) \div (-1) = 1 ).
Answer: ( 1 ).

2. Verify that ( a \div (b + c) \neq (a \div b) + (a \div c) ) for each of the following values of ( a, b, c ):

(a) ( a = 12, b = -4, c = 2 )

Left side: ( b + c = -4 + 2 = -2 ), so ( 12 \div (-2) = -6 ).
Right side: ( 12 \div (-4) = -3 ), ( 12 \div 2 = 6 ), so ( (-3) + 6 = 3 ).
Since ( -6 \neq 3 ), division is not distributive over addition (section 1.5).
Answer: ( 12 \div (-2) = -6 \neq 3 = (-3) + 6 ).

(b) ( a = -10, b = 1, c = 1 )

Left side: ( b + c = 1 + 1 = 2 ), so ( (-10) \div 2 = -5 ).
Right side: ( (-10) \div 1 = -10 ), ( (-10) \div 1 = -10 ), so ( (-10) + (-10) = -20 ).
Since ( -5 \neq -20 ), the statement is verified.
Answer: ( (-10) \div 2 = -5 \neq -20 = (-10) + (-10) ).

3. Fill in the blanks:

(a) ( 369 \div __ = 369 )

( a \div b = a ) implies ( b = 1 ) (section 1.5).
So, ( 369 \div 1 = 369 ).
Answer: ( 1 ).

(b) ( (-75) \div __ = -1 )

Let the blank be ( x ). Then, ( (-75) \div x = -1 ).
So, ( x = (-75) \div (-1) = 75 ).
Answer: ( 75 ).

(c) ( (-206) \div __ = 1 )

( (-206) \div x = 1 ).
So, ( x = (-206) \div 1 = -206 ).
Answer: ( -206 ).

(d) ( -87 \div __ = 87 )

( (-87) \div x = 87 ).
So, ( x = (-87) \div 87 = -1 ).
Answer: ( -1 ).

(e) ( __ \div 1 = -87 )

( x \div 1 = x ), so ( x = -87 ).
Answer: ( -87 ).

(f) ( __ \div 48 = -1 )

( x \div 48 = -1 ).
So, ( x = -1 \times 48 = -48 ).
Answer: ( -48 ).

(g) ( 20 \div __ = -2 )

( 20 \div x = -2 ).
So, ( x = 20 \div (-2) = -10 ).
Answer: ( -10 ).

(h) ( __ \div 4 = -3 )

( x \div 4 = -3 ).
So, ( x = -3 \times 4 = -12 ).
Answer: ( -12 ).

4. Write five pairs of integers ( (a, b) ) such that ( a \div b = -3 ). One such pair is ( (6, -2) ).

We need ( a \div b = -3 ), so ( a = -3b ).
Pairs:
1. ( (6, -2) ): ( 6 \div (-2) = -3 ) (given).
2. ( (-6, 2) ): ( (-6) \div 2 = -3 ).
3. ( (9, -3) ): ( 9 \div (-3) = -3 ).
4. ( (-9, 3) ): ( (-9) \div 3 = -3 ).
5. ( (12, -4) ): ( 12 \div (-4) = -3 ).
Answer: ( (6, -2), (-6, 2), (9, -3), (-9, 3), (12, -4) ).

5. The temperature at 12 noon was ( 10^\circ \text{C} ) above zero. If it decreases at the rate of ( 2^\circ \text{C} ) per hour until midnight, at what time would the temperature be ( 8^\circ \text{C} ) below zero? What would be the temperature at midnight?

Initial temperature: ( 10^\circ \text{C} ).
Rate of decrease: ( 2^\circ \text{C} ) per hour.
Time from 12 noon to midnight: 12 hours.
Time to reach ( -8^\circ \text{C} ):
- Target temperature: ( -8^\circ \text{C} ).
- Total decrease needed: ( 10 - (-8) = 18^\circ \text{C} ).
- Time required: ( 18 \div 2 = 9 ) hours.
- Time: 12 noon + 9 hours = 9:00 PM.
Temperature at midnight:
- Total decrease: ( 12 \times 2 = 24^\circ \text{C} ).
- Temperature: ( 10 - 24 = -14^\circ \text{C} ).
Answer: The temperature reaches ( -8^\circ \text{C} ) at 9:00 PM. At midnight, the temperature is ( -14^\circ \text{C} ).

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?

Marks for correct answers: ( 12 \times 3 = 36 ).
Total score: 20.
Marks from incorrect answers: ( 20 - 36 = -16 ).
Marks per incorrect answer: ( -2 ).
Number of incorrect answers: ( (-16) \div (-2) = 8 ).
Answer: 8 incorrect answers.

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

Marks for correct answers: ( 7 \times 3 = 21 ).
Total score: -5.
Marks from incorrect answers: ( -5 - 21 = -26 ).
Marks per incorrect answer: ( -2 ).
Number of incorrect answers: ( (-26) \div (-2) = 13 ).
Answer: 13 incorrect answers.

7. An elevator descends into a mine shaft at the rate of ( 6 , \text{m/min} ). If the descent starts from ( 10 , \text{m} ) above the ground level, how long will it take to reach ( -350 , \text{m} )?

Initial position: ( 10 , \text{m} ).
Final position: ( -350 , \text{m} ).
Total distance: ( 10 - (-350) = 360 , \text{m} ).
Rate: ( 6 , \text{m/min} ).
Time: ( 360 \div 6 = 60 ) minutes = 1 hour.
Answer: 1 hour.

Other exercises.

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