NCERT Solutions for Class 7th Maths Chapter 2 Math Talk — Multiplication of Integers

Book page 31 Updated on2026-09-19

Q1.

Consider the numbers represented by the following tokens: (a), (b), (c). We can see that all of them represent the number (– 2). Now, take 4 times each of these token sets. That is, place each set into the empty bag 4 times. What integer do we get as the final answer in each case? Do we get different answers because the sets look different, or the same answer because they all represent – 2?

Answer

All three give the same answer, –8. The sets look different, but they are worth the same.

First check what each set is worth.

Set	Tokens shown	Value
(a)	2 reds	–2
(b)	4 reds and 2 greens	–4 + 2 = –2
(c)	6 reds and 4 greens	–6 + 4 = –2

Now place each set into an empty bag 4 times.

(a) 4 × 2 reds = 8 reds → –8
(b) 4 × (4 reds + 2 greens) = 16 reds + 8 greens; cancel 8 zero pairs → 8 reds → –8
(c) 4 × (6 reds + 4 greens) = 24 reds + 16 greens; cancel 16 zero pairs → 8 reds → –8

Why it happens: sets (b) and (c) are just set (a) with extra zero pairs thrown in — one extra pair in (b), two in (c). Repeating a set 4 times repeats its zero pairs 4 times as well, and those cancel out at the end. So the answer depends only on the value of the set, never on how many tokens it happens to contain.

Tip: this is an important check on any model. If two pictures stand for the same number, they must behave identically in every calculation — otherwise the model would not be describing numbers at all.

Q2.

Check this for 5 × 4, by taking different token sets corresponding to 4.

Answer

Every token set worth 4 gives the same product, 20.

Set worth 4	Placed 5 times	After cancelling zero pairs
4 greens	20 greens	20
5 greens + 1 red	25 greens + 5 reds	20
6 greens + 2 reds	30 greens + 10 reds	20
10 greens + 6 reds	50 greens + 30 reds	20

Take 6 greens + 2 reds (value 6 – 2 = 4)
5 times: 30 greens and 10 reds
Cancel 10 zero pairs
20 greens left = 20

Why it happens: each extra zero pair inside the set is copied 5 times, giving 5 extra zero pairs at the end — and 5 zero pairs are still worth 0. The padding is copied faithfully and then cancels itself, so it can never affect the product.

Q3.

We have seen that – 4 × 2 is the number obtained by removing 2 positive tokens from the empty bag 4 times. We know that removing or subtracting a number is the same as adding its inverse. Using this, can – 4 × 2 be defined through a process of addition of tokens instead of removal of tokens?

Answer

Yes. Removing 2 positives is the same as adding 2 negatives.

Remove 2 greens, 4 times
= Add 2 reds, 4 times
= 8 reds
(–4) × 2 = –8

Both routes empty into the same bag, so both descriptions of (–4) × 2 are correct.

Why it happens: taking one green out lowers the value by 1, and putting one red in also lowers the value by 1. The two moves have identical effects, so any sequence of removals can be rewritten as a sequence of additions of the opposite colour. This is the token version of the rule a – b = a + (–b).

Tip: the addition form is easier to work with, because you never have to stock the bag with zero pairs first. It also explains the neat statement (–a) × b = –(a × b).

Was this helpful?