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NCERT Solutions for Class 7th Maths Chapter 2 In-text Questions — Multiplication of Integers

Book page 34–35 Updated on2026-09-19

Q1.
Consider the expression 1 × a. We know that the value of this expression is ‘a’ for all positive integers. Is this true for all negative integers too?
Answer

Yes. 1 × a = a for every integer a, positive or negative.

Use the token model. The multiplier is 1, so you place the multiplicand into the empty bag just once.

If a = –5, place 5 reds in the bag, once
The bag holds 5 reds
1 × (–5) = –5

So in general,

1 × a = a  (for all integers a)
Why it happens: doing something once changes nothing about it. The bag ends up holding precisely the set you put in, whatever colour those tokens were. That is why 1 is called the multiplicative identity, and why the sign of a survives untouched.
Q2.
What is the value of the expression – 1 × a?
Answer

–1 × a = –a — the additive inverse of a — for every integer a.

a–1 × aWhat happened
7–7Same magnitude, sign flipped
–77Same magnitude, sign flipped
000 is its own inverse
When a is positive, the product has magnitude a and is negative
When a is negative, the product has magnitude a and is positive
In both cases the product is the additive inverse of a
–1 × a = –a
Why it happens: the multiplier –1 says “remove the multiplicand once”. Removing a from an empty bag leaves the bag owing a, and owing a is exactly –a. So multiplying by –1 is the same as taking the additive inverse — it is a sign-flipper, nothing more.
Tip: this single fact settles a whole family of questions. For example, the number whose product with –1 is –31 must be 31, because flipping the sign of 31 gives –31.
Q3.
In the case of integers, is the product the same when we swap the multiplier and the multiplicand? Try this for some numbers.
Answer

Yes. Swapping them never changes the product.

3 × (–4) = –12  and  (–4) × 3 = –12
(–15) × (–8) = 120  and  (–8) × (–15) = 120
7 × (–9) = –63  and  (–9) × 7 = –63

The two sides of each pair agree every time.

Why it happens: the token stories are different but the bags end up the same. For 3 × (–4) you put 4 reds in three times; for (–4) × 3 you take 3 greens out four times. The first leaves 12 reds, the second leaves 12 reds. Different instructions, identical bag.
Q4.
Observe the following pairs of multiplications (fill in the blanks where needed): 3 × – 4 = –12, – 4 × 3 = –12; – 30 × 12 = _______, 12 × – 30 = _______; –15 × – 8 = 120, – 8 × –15 = 120; 14 × – 5 = –70, – 5 × _____ = – 70. What do you notice in these pairs of multiplication statements?
Answer

The filled table is:

First statementSwapped statement
3 × (–4) = –12(–4) × 3 = –12
(–30) × 12 = –36012 × (–30) = –360
(–15) × (–8) = 120(–8) × (–15) = 120
14 × (–5) = –70(–5) × 14 = –70
(–30) × 12: 30 × 12 = 360, one negative → –360
12 × (–30): same magnitudes, one negative → –360
(–5) × ___ = –70: 70 ÷ 5 = 14, and one negative is already there → 14

What we notice: the product is the same when the multiplier and multiplicand are swapped.

Why it happens: a product carries two pieces of information — a magnitude and a sign — and swapping disturbs neither. The magnitude is the product of the two magnitudes, and that is unchanged because whole-number multiplication is commutative. The sign depends only on how many of the two numbers are negative, and swapping does not change that count.
Q5.
The magnitude of the product does not change when the multiplier and the multiplicand are swapped… Will this always happen?
Answer

Yes, always. The magnitude of a product depends only on the two magnitudes.

magnitude of (a × b) = (magnitude of a) × (magnitude of b)
and for positive numbers, m × n = n × m
so the magnitude cannot change when a and b are swapped

For example, both (–15) × (–8) and (–8) × (–15) have magnitude 15 × 8 = 120.

Why it happens: the sign of each number tells us the direction of the action, not its size. Once the signs are set aside, only 15 and 8 are left, and a 15-by-8 array of objects is the same array as an 8-by-15 one, just turned on its side. That old fact about positive numbers is doing all the work here.
Q6.
Does the sign of the product change if we swap the multiplier and multiplicand?
Answer

No. Whatever the signs are, they give the same result before and after swapping.

  • If both are positive, the product is positive before and after the swap.
  • If both are negative, the product is positive before and after the swap.
  • If one is positive and the other negative, the product is negative before and after the swap — swapping only decides which of the two is written first.

So neither the magnitude nor the sign changes, and therefore

a × b = b × a  for any two integers a and b

That is, multiplication is commutative for integers.

Why it happens: the sign rule asks only one question — are the two signs the same or different? “Same” and “different” are relationships between the two numbers, and a relationship does not care which number you name first. That is why the sign is untouched by swapping.
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