Ex 1.4, Mathematics Class 7 Integers Chapter 1 : NCERT Solution
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
= -3
When we divide a negative integer by a positive integer, we first divide them as whole numbers and then put minus sign (-) before the quotient.
(b) 50 ÷ (-5)
= -10
When we divide a negative integer by a positive integer, we first divide them as whole numbers and then put minus sign (-) before the quotient.
(c) (-36) ÷ (-9)
= 4
When we divide a negative integer by a negatve integer, we first divide them as a whole numbers and then put positive sign (+) before the quotient.
(d) (-49) ÷ (49)
= -1
When we divide a negative integer by a positive integer, we first divide them as whole numbers and then put minus sign (-) before the quotient.
(e) 13 ÷ [(-2) + 1]
= 13 ÷ [-1]
= -13
When we divide a negative integer by a positive integer, we first divide them as whole numbers and then put minus sign (-) before the quotient.
(f) 0 ÷ (-12)
= 0
When we divide zero by a negative integer, it give zero.
(g) (-31) ÷ [(-30) + (-1)]
= -31 ÷ [-30 -1]
= -31 ÷ [-31]
= 1
When we divide a negative integer by a negative integer, we first divide them as a whole numbers and then put positive sign (+) before the quotient.
(h) [(-36) ÷ 12] ÷ 3
= [-3] ÷ 3
= -1
When we divide negative integer by a positive integer, we first divide them as a whole numbers and then put minus sign (-) before the quotient.
(i) [(-6) + 5)] ÷ [(-2) + 1]
= [-1] ÷ [-1]
= 1
When we divide negative integer by a negative integer, we first divide them as a whole numbers and then put positive sign (+) before the quotient.
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 :
(a) a = 12, b = -4, c = 2
a ÷ (b + c) ≠ (a ÷ b) + ( a ÷ c)
12 ÷ (-4 + 2) ≠ (12 ÷ (-4) + (12 ÷ 2)
12 ÷ (-2) ≠ (-3) + 6
-6 ≠ 3
(b) a = (-10), b = 1, c = 1
a ÷ (b + c) ≠ (a ÷ b) + ( a ÷ c)
-10 ÷ (1 + 1) ≠ (-10 ÷ 1) + (-10 ÷ 1)
-10 ÷ 2 ≠ (-10) + (-10)
-5 ≠ -20
3. 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
Solution :
(a) Let us assume the missing integer be x,
Then,
= 369 ÷ x = 369
= x = 369/369
= x = 1
(b) Let us asume the missing integer be x,
Then,
(-75) ÷ x = -1
x = -75/-1
x = 75
(c) Let us assume the missing integer be x,
Then,
(-206) ÷ x = 1
x = -206/1
x = -206
(d) Let us assume the missing integer be x,
Then,
-87 ÷ x = 87
x = -87/87
x = -1
(e) Let us assume the missing integer be x,
Then,
x ÷ 1 = -87
x = -87 ✕ 1
x = -87
(f) Let us assume the missing integer be x,
Then,
x ÷ 48 = -1
x = -1 ✕ 48
x = -48
(g) Let us assume the missing integer be x,
Then,
20 ÷ x = -2
x = (20)/ (-2)
x = -10
(h) Let us assume the missing integer be x,
Then,
x ÷ 4 = -3
x = -3 ✕ 4
x = -12
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℃ above zero. If it decreases at the rate of 2℃ per hour until midnight, at what time would be temperature be 8℃ below zero ? What would be the temperature at mid night ?
Solution :
Given :-
Temperature at the beginning i.e., at 12 noon = 10℃
Rate of change of temperature = -2℃ per hour
Then,
Temperature at 1 PM
= 10 + (-2)
= 10 - 2
= 8℃
Temperature at 2 PM
= 8 + (-2)
= 8 - 2
= 6℃
Temperature at 3 PM
= 6 + (-2)
= 6 - 2
= 4℃
Temperature at 4 PM
= 4 + (-2)
= 4 - 2
= 2℃
Temperature at 5 PM
= 2 + (-2)
= 2 - 2
= 0℃
Temperature at 6 PM
= 0 + (-2)
= 0 - 2
= -2℃
Temperature at 7 PM
= -2 + (-2)
= -2 - 2
= -4℃
Temperature at 8 PM
= -4 + (-2)
= -4 -2
= -6℃
Temperature at 9 PM
= -6 + (-2)
= -6 -2
= -8℃
∴ At 9 PM the temperature will be 8℃ below zero
Then,
The tempeature at mid night i.e., at 12 AM
Change in temperature in 12 hours
= -2℃ ✕ 12
= -24℃
So, at midnight temperature will be
= 10 + (-24)
= 10 - 24
= -14℃
So, at midnight temperature will be 14℃ 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 :
According to question,
marks awarded for 1 correct answer = +3
marks awarded for 1 wrong answer = -2
(i) Radhika scored 20 marks
Then,
Total marks awarded for 12 correct answers
= 12 ✕ 3
= 36
marks awarded for incorrect answers
= Total score - Total marks awarded for 12 correct answers
= 20 - 36
= -16
So, the number of incorrect answers made by Radhika
= (-16) ÷ (-2)
= 8
(ii) Mohan scored -5 marks
Then,
Total marks awarded for 7 correct answers
= 7 ✕ 3
= 21
Marks awarded for incorrect answers
= Total score - total marks awarded for 12 correct 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 6m/min. If the descent starts from 10 m above the ground level, how long will it take to reach -350m.
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 -6m
= 1 min
So, time taken by the elevator to descend -360m
= (-360) ÷ (-60)
= 60 minutes
= 1 hour