NCERT Solutions for Class 7 maths Chapter 10: Operations with Integers

Using the token interpretation, find the values of:
(a) 3 × (-2)
(b) (-5) × (-2)
(c) (-4) × (-1)
(d) (-7) × 3

If 123 × 456 = 56088, without calculating, find the value of:
(a) (-123) × 456
(b) (-123) × (-456)
(c) (123) × (-456)
Solution:
(a) Given 123 × 456 = 56088 …..(i)
Then(a) (-123) × 456
This is the product of a negative and a positive integer, so the result will be negative.
i.e. -(123) × 456 = -(123 × 456) = -56088

Try to frame a simple rule to multiply two integers.
Solution:
Rule for multiplying two integers:

Find the following products.
(a) 4 × (-3)
(b) (-6) × (-3)
(c) (-5) × (-1)
(d) (-8) × 4
(e) (-9) × 10
(f) 10 × (-17)
Solution:
(a) Here, 4 × (-3) = – [(4) × (3)] = -12
(∵ Multiplier is positive and the multiplicand is negative, their product is negative)

Find the values of:
(a) 14 × (-15)
(b) -16 × (-5)
(c) 36 ÷ (-18)
(d) (-46) ÷ (-23)
Solution:
(a) To find the value of 14 × (-15), we multiply the numbers and apply the rule that a positive number multiplied by a negative number results in a negative number.
∴ 14 × (-15) = -(14 × 15) = -210

A freezing process requires that the room temperature he lowered from 32°C at the rate of 5°C every hour. What will be the room temperature 10 hours after the process begins?
Solution:
The temperature is lowered at the rate of 5°C every hour for 10 hours.
Total temperature drop = 5°C/hour × 10 hour = 50°C
The initial room temperature is 32°C.
To find the final temperature, we have to subtract the initial temperature from the total temperature drop.
∴ Final temperature = 32 – 50 = -18°C

A cement company earns a profit of ₹ 8 per bag of white cement sold and a loss of ₹ 5 per bag of grey cement sold. [Represent the profit/loss as integers.]
(a) The company sells 3,000 bags of white cement and 5,000 bags of grey cement in a month. What is its profit or loss?
(b) If the number of bags of grey cement sold is 6,400 bags, what is the number of bags of white cement the company must sell to have neither profit nor loss?
Solution:
(a) Profit on one white cement bag = ₹ 8
Loss on one grey cement bag = ₹ 5
Profit from selling white cement = 3000 bags × ₹ 8/bag = ₹ 24000
Loss from selling white cement = 5000 bags × ₹ 5/bag = ₹ 25,000
[∵ loss is more than profit, i.e., loss]
Total loss = 25,000 – 24,000 = 1,000
∴ The company has a loss of ₹ 1000.

Replace the blank with an integer to make a true statement.
(a) (-3) × ________ = 27
(b) 5 × ________ = (-35)
(c) ________ × (-8) = (-56)
(d) ________ × (-12) = 132
(e) ________ ÷ (-8) = 7
(f) ________ ÷ 12 = -11
Solution:
(a) Let -3 × x = 27
⇒ x = = -9

Find the values of the following expressions:
(a) (-5) × (18 + (-3))
(b) (-7) × 4 × (-1)
(c) (-2) × (-1) × (-5) × (-3)
Solution:
(a) Given (-5) × [(18 + (-3)] = -5 × 18 + (-5) × (-3) [∵ a(b + c) = ab + ac]
= -90 + 15
= -75

Find the values of the following expressions:
(a) (-27) ÷ 9
(b) 84 ÷ (-4)
(c) (-56) ÷ (-2)
Solution:
(a) (-27) ÷ 9 or 9 × ________ = -27
We know that, 9 × (-3) = -27
So, (-27) ÷ 9 = -3

Find the integer whose product with (-1) is:
(a) 27
(b) -31
(c) -1
(d) 1
(e) 0
Solution:
(a) Here, (-1) × (-27) = 27
∴ -27 is the required integer.

If 47 – 56 + 14 – 8 + 2 – 8 + 5 = -4, then find the value of -47 + 56 – 14 + 8 – 2 + 8 – 5 without calculating the full expression.
Solution:
Given expression is 47 – 56 + 14 – 8 + 2 – 8 + 5 = -4 = E1 (Let)
Here we see that the second expression is the negative of the first expression, then
E2 = -47 + 56 – 14 + 8 – 2 + 8 – 5
= -1(47 – 56 + 14 – 8 + 2 – 8 + 5) (Taking -1 common)
= -1(68 – 72)
= -1(-4)
= 4

Do you remember the Collatz Conjecture from last year? Try a modified version with integers. The rule is — start with any number; if the number is even, take half of it; if the number is odd, multiply it by -3 and add 1; repeat. An example sequence is shown below.

Try this with different starting numbers: (-21), (-6), and so on. Describe the patterns you observe.
Solution:
(a) Following the rules for the sequence starting with -7.
The rule is if even, take half, if odd, multiply by -3, and add 1.
Start with -7 (odd)
[(-7) × (-3)] + 1 = 21 + 1 = 22 (even)
then 22 ÷ 2 = 11 (odd)
then 11 × (-3) + 1 = -33 + 1 = -32 (even)
then (-32) ÷ 2 = -16 (even)
then (-16) ÷ 2 = -8 (even)
then (-8) ÷ 2 = -4 (even)
then (-4) ÷ 2 = -2 (even)
then (-2) ÷ 2 = – 1 (odd)
then [(-1) × (-3)] + 1 = 4 (even)
then 4 ÷ 2 = 2 (even)
then 2 ÷ 2 = 1 (odd)

In a test, (+4) marks are given for every correct answer and (-2) marks are given for every incorrect answer.
(a) Anita answered all the questions in the test. She scored 40 marks even though 15 of her answers were correct. How many of her answers were incorrect? How many questions are in the test?
(b) Anil scored (-10) marks even though he had 5 correct answers. How many of his answers were incorrect? Did he leave any questions unanswered?
Solution:
(a) Anita had 15 correct answers, and each correct answer is worth +4 marks.
Marks from correct answers = 15 × 4 = 60
∴ Anita total score = 40
Marks lost = 60 – 40 = 20
Each incorrect answer is worth -2 marks
∴ No. of incorrect answers = = 10
Anita had 10 incorrect answers.
Total no. of questions = sum of correct and incorrect answers
= 15 + 10
= 25
Hence, Anita had 10 incorrect answers out of 25 questions on the test.

Pick the pattern – find the operations done by the machine shown below:

Solution:
Operation done by machine is (First Number) – [Second number × Third Number]
I row: 4 – [(8) × -3] = 4 – [-24] = 4 + 24 = 28.
II row: 6 – [9 × 6] = 6 – [54] = 6 – 54 = -48.
III row: 2 – [3 × (-2)] = 2 – (-6) = 2 + 6 = 8.
IV row: -9 – [5 × (-8)] = -9 – (-40) = -9 + 40 = 31.
V row: 7 – (-4 × -6) = 7 – (24) = 7 – 24 = -17.
VI row: -16 – (-6 × -9) = -16 – (54) = -70.
Thus, we have found the pattern that satisfies all the values.
Hence, the missing operation is -70.

Imagine you’re in a place where the temperature drops by 5°C each hour. If the temperature is currently at 8°C, write an expression that denotes the temperature after 4 hours.
Solution:
Current temperature = 8°C
The temperature drops by 5°C each hour
Total drop in temperature = Temperature drop in 1 hour × no. of hours = 5 × 4
Temperature after 4 hours = 8 – 4 × 5
Hence required expression is 8 – (5 × 4).

Find 3 consecutive numbers with a product of (a) -6, (b) 120.
Solution:
(a) Let three consecutive numbers be n – 1, n, and n + 1.
Then product = (n – 1) (n) (n + 1) = -6
The only consecutive integers whose product is -6 are -2, -1, and 0.
But -2 × -1 × o = 0, not possible.
Consecutive numbers are integers.
The product is negative, so there must be an odd number of negative integers.
Integers are -2, -1, 1.
Their product = – 2 × -1 × 1 = 2, not possible
Integers are -3, -2, -1.
Their product = [-3 × -2] × -1 = 6 × -1 = -6
Hence, consecutive integers are -3, -2, -1.

An alien society uses a peculiar currency called ‘pibs’ with just two denominations of coins — a + 13 pibs coin and a – 9 pibs coin. You have several of these coins. Is it possible to purchase an item that costs +85 pibs? Yes, we can use 10 coins of +13 pibs and 5 coins of -9 pibs to make a total of +85. Using the two denominations, try to get the following totals:
(a) +20
(b) +40
(c) -50
(d) +8
(e) +10
(f) -2
(g) +1
[Hint: Writing down a few multiples of 13 and 9 can help.]
(h) Is it possible to purchase an item that costs 1568 pibs?
Solution:
The currency has two denominations: +13 pibs and -9 pibs.
We need to determine if it is possible to make the given totals.
This is a linear equation of the form 13x – 9y = total,
where x and y are non-negative integers.
(a) Now total = +20
Then we need to find integers 13x – 9y = 20, x, y > 0 such that
If x = 1, then 13 – 9y = 20 ⇒ -9y = 7. No solution.
If x = 2, then 26 – 9y = 20 ⇒ -9y = -6. No solution.
IF x = 3 then 39 – 9y = 20 ⇒ -9y = -19. No solution.
If x = 4, then 52 – 9y = 20 ⇒ -9y = -32. No solution.
If x = 5 then 65 – 9y = 20 ⇒ -9y = -45 ⇒ y = 5
Hence, x = 5, y = 5 is a valid solution.

Find the values of:
(a) (32 × (-18)) ÷ ((-36))
(b) (32) ÷ ((-36) × (-18))
(c) (25 × (-12)) ÷ ((45) × (-27))
(d) (280 × (-7)) ÷ ((-8) × (-35))
Solution:
(a) Here 32 × (-18) ÷ (-36)
= -576 ÷ (-36)
= 16

Arrange the expressions given below in increasing order:
(a) (-348) + (-1064)
(b) (-348) – (-1064)
(c) 348 – (-1064)
(d) (-348) × (-1064)
(e) 348 × (-1064)
(f) 348 × 964
Solution:
(a) (-348) + (-1064)
Add two negative numbers → keep negative, add absolute values:
-348 + (-1064) = -1412.

Given that (-548) × 972 = -532656, write the values of:
(a) (-547) × 972
(b) (-548) × 971
(c) (-547) × 971
Solution:
(a) Given: (-547) × 972 = -532656
We need to find (-547) × 972 = (-548 + 1) × 972
= -548 × 972 + 972
= -532656 + 972
= -531684

Given that 207 × (-33 + 7) = -5382, write the value of -207 × (33 – 7) = ________
Solution:
Here 207 × (-33 + 7)
= 207 × (-26)
= -5382
Now -207 × (33 – 7)
= -207 × 26
= -(207 × 26)
= -5382

Use the numbers 3, -2, 5, -6 exactly once and the operations ‘+’, ‘-’, and ‘×’ exactly once and brackets as necessary to write an expression such that-
(a) The result is the maximum possible
(b) The result is the minimum possible
Solution:
(a) Maximum possible value = [(-2) – (3 + 5)] × (-6)
= (-2 – 8) × (-6)
= 60

Fill in the blanks in at least 5 different ways with integers:

Solution:
(a) Here

(i) 0 + -6 × 6 = -36
(ii) 4 + (-5 × 8) = -36
(iii) -4 + -8 × 4 = -36
(iv) 12 + -8 × 6 = -36
(v) -3 + 3 × (-11) = -36