NCERT Solutions Class 7 Maths Chapter 1 Integers Ex 1.4

NCERT Solutions Class 7 Maths
Chapter – 1 (Integers)

The NCERT Solutions in English Language for Class 7 Mathematics Chapter – 1 Integers Exercise 1.4 has been provided here to help the students in solving the questions from this exercise.

Chapter 1: Integers

Exercise – 1.4

1. 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]

Solution –
(a) (–30) ÷ 10
= (–30) ÷ 10
= – 3

(b) 50 ÷ (–5)
= (50) ÷ (-5)
= – 10

(c) (–36) ÷ (–9)
= (-36) ÷ (-9)
= 4

(d) (– 49) ÷ (49)
= (–49) ÷ 49
= – 1

(e) 13 ÷ [(–2) + 1]
= 13 ÷ [(–2) + 1]
= 13 ÷ (-1)
= – 13

(f) 0 ÷ (–12)
= 0 ÷ (-12)
= 0
When we divide zero by a negative integer gives zero.

(g) (–31) ÷ [(–30) + (–1)]
= (–31) ÷ [(–30) + (–1)]
= (–31) ÷ [– 30 – 1]
= (–31) ÷ (–31)
= 1

(h) [(–36) ÷ 12] ÷ 3
First we have to solve the integers with in the bracket,
= [(–36) ÷ 12]
= (–36) ÷ 12
= – 3
Then,
= (-3) ÷ 3
= -1

(i) [(– 6) + 5)] ÷ [(–2) + 1]
= [-1] ÷ [-1]
= 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 –
From the question, a ÷ (b + c) ≠ (a ÷ b) + (a ÷ c)
Given, a = 12, b = – 4, c = 2
Now, consider LHS = a ÷ (b + c)
= 12 ÷ (-4 + 2)
= 12 ÷ (-2)
= -6
Then, consider RHS = (a ÷ b) + (a ÷ c)
= (12 ÷ (-4)) + (12 ÷ 2)
= (-3) + (6)
= 3
By comparing LHS and RHS
= -6 ≠ 3
= LHS ≠ RHS
Hence, the given values are verified.

(b) a = (–10), b = 1, c = 1
From the question, a ÷ (b + c) ≠ (a ÷ b) + (a ÷ c)
Given, a = (-10), b = 1, c = 1
Now, consider LHS = a ÷ (b + c)
= (-10) ÷ (1 + 1)
= (-10) ÷ (2)
= -5
Then, consider RHS = (a ÷ b) + (a ÷ c)
= ((-10) ÷ (1)) + ((-10) ÷ 1)
= (-10) + (-10)
= -10 – 10
= -20
By comparing LHS and RHS
= -5 ≠ -20
= LHS ≠ RHS
Hence, the given values are verified.

3. Fill in the blanks:
(a) 369 ÷ ___ = 369
(b) (-75) ÷ ___ = -1
(c) (-206) ÷ ___ =1
(d) -87 ÷ ___ = -87
(e) ___ ÷ 1 = -87
(g) 20 ÷ ___ = -2

Solution –
(a) 369 ÷ ___ = 369
Let us assume the missing integer be x,
Then,
= 369 ÷ x = 369
= x = $\frac{369}{369}$
= x = 1
Now, put the valve of x in the blank.
= 369 ÷ 1 = 369

(b) (–75) ÷ _____ = –1
Let us assume the missing integer be x,
Then,
= (-75) ÷ x = -1
= x = $\frac{-75}{-1}$
= x = 75
Now, put the valve of x in the blank.
= (-75) ÷ 75 = -1

(c) (–206) ÷ _____ = 1
Let us assume the missing integer be x,
Then,
= (-206) ÷ x = 1
= x = $\frac{-206}{1}$
= x = -206
Now, put the valve of x in the blank.
= (-206) ÷ (-206) = 1

(d) – 87 ÷ _____ = 87
Let us assume the missing integer be x,
Then,
= (-87) ÷ x = 87
= x = $\frac{-87}{87}$
= x = -1
Now, put the valve of x in the blank.
= (-87) ÷ (-1) = 87

(e) _____ ÷ 1 = – 87
Let us assume the missing integer be x,
Then,
= (x) ÷ 1 = -87
= x = (-87) × 1
= x = -87
Now, put the valve of x in the blank.
= (-87) ÷ 1 = -87

(f) _____ ÷ 48 = –1
Let us assume the missing integer be x,
Then,
= (x) ÷ 48 = -1
= x = (-1) × 48
= x = -48
Now, put the valve of x in the blank.
= (-48) ÷ 48 = -1

(g) 20 ÷ _____ = –2
Let us assume the missing integer be x,
Then,
= 20 ÷ x = -2
= x = $\frac{20}{-2}$
= x = -10
Now, put the valve of x in the blank.
= (20) ÷ (-10) = -2

(h) _____ ÷ (4) = –3
Let us assume the missing integer be x,
Then,
= (x) ÷ 4 = -3
= x = (-3) × 4
= x = -12
Now, put the valve of x in the blank.
= (-12) ÷ 4 = -3

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

Solution:-
(i) (15, -5) Because, 15 ÷ (–5) = (–3)
(ii) (-15, 5) Because, (-15) ÷ (5) = (–3)
(iii) (18, -6) Because, 18 ÷ (–6) = (–3)
(iv) (-18, 6) Because, (-18) ÷ 6 = (–3)
(v) (21, -7) Because, 21 ÷ (–7) = (–3)

5. The temperature at 12 noon was 10^oC above zero. If it decreases at the rate of 2^oC 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 was 10°C above zero i.e. +10°C
Rate of decrease in temperature per hour = 2°C
Number of hours from 12 noon to midnight = 12
∴ Change in temperature in 12 hours = 12 × (-2°C) = -24°C
∴ Temperature at midnight = +10°C + (-24°C) = -14°C
Hence, the required temperature at midnight = -14°C
Difference in temperature between + 10°C and -8°C = +10°C – (-8°C)
= +10°C + 8°C = 18°C
Number of hours required = $\frac{18^{\circ}C}{2^{\circ}C}$
= 9 hours
∴ Time after 9 hours from 12 noon = 9 pm.

Temperature at the beginning i.e., at 12 noon = 10^oC
Rate of change of temperature = – 2^oC per hour
Then,
Temperature at 1 PM = 10 + (-2) = 10 – 2 = 8^oC
Temperature at 2 PM = 8 + (-2) = 8 – 2 = 6^oC
Temperature at 3 PM = 6 + (-2) = 6 – 2 = 4^oC
Temperature at 4 PM = 4 + (-2) = 4 – 2 = 2^oC
Temperature at 5 PM = 2 + (-2) = 2 – 2 = 0^oC
Temperature at 6 PM = 0 + (-2) = 0 – 2 = -2^oC
Temperature at 7 PM = -2 + (-2) = -2 -2 = -4^oC
Temperature at 8 PM = -4 + (-2) = -4 – 2 = -6^oC
Temperature at 9 PM = -6 + (-2) = -6 – 2 = -8^oC
∴ at 9 PM the temperature will be 8^oC below zero
Then,
The temperature at mid-night i.e., at 12 AM
Change in temperature in 12 hours = -2^oC × 12 = – 24^oC
So, at midnight temperature will be = 10 + (-24) = – 14^oC
So, at midnight temperature will be 14^oC below 0.

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 –
From the question,
Marks awarded for 1 correct answer = + 3
Marks awarded for 1 wrong answer = -2
(i) Radhika scored 20 marks
Then,
Marks obtained by Radhika for 12 correct answers = (+3) × 12 = 36
Marks obtained by Radhika for correct answers = 12 × 3 = 36
Marks awarded for incorrect answers = Total score – Total marks awarded for 12 correct
∴ Marks obtained by Radhika for incorrect answers = 20 – 36 = -16
So, the number of incorrect answers made by Radhika = (-16) ÷ (-2) = 8

(ii) Mohini scored -5 marks
Then,
Marks scored by Mohini = -5
Number of correct answers = 7
∴ Marks obtained by Mohini for 7 correct answers = 7 × (+3) – 21
Marks awarded for incorrect answers = Total score – Total marks awarded for 12 correct
Marks obtained for incorrect answers = – 5 – 21 = (– 26)
So, the number of incorrect answers made by Mohini = (-26) ÷ (-2) = 13

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 –
From the question,
The initial height of the elevator = 10 m
Final depth of elevator = – 350 m [∵ distance descended is denoted by a negative integer]
The total distance to descended by the elevator = (-350) – (10) = – 360 m
Then,
Time taken by the elevator to descend -6 m = 1 min
So, time taken by the elevator to descend – 360 m = (-360) ÷ (-60)
= 60 minutes = 1 hour
Hence, the required time = 1 hour.