NCERT Solutions Ganita Prakash (Part 1) Chapter 2 In-text Questions — Swapping and Grouping

Book page 31 Updated on2026-09-05

Q1.

Does adding the terms of an expression in any order give the same value? Take some more expressions and check. Consider expressions with more than 3 terms also.

Answer

Yes. Any order and any grouping gives the same value.

Three terms: 9 + 4 + 6

(9 + 4) + 6 = 13 + 6 = 19
9 + (4 + 6) = 9 + 10 = 19
6 + 9 + 4 = 15 + 4 = 19

Four terms (one negative): 15 + (–8) + 5 + 12

(15 + (–8)) + 5 + 12 = 7 + 5 + 12 = 24
15 + 5 + 12 + (–8) = 32 + (–8) = 24
(15 + 5) + (12 + (–8)) = 20 + 4 = 24

Five terms: 20 + (–7) + (–3) + 11 + (–1) = 20 in every order.

Why it happens: Commutativity lets us swap any two terms and associativity lets us bracket any two together. Repeating these two moves turns any order into any other order, so all orders give the same sum.

Tip: Use this to add faster. In 20 + (–7) + (–3) + 11 + (–1), first collect all the negatives: –7 – 3 – 1 = –11, then 20 + 11 – 11 = 20.

Q2.

Can you explain why this is happening using the Token Model of integers that we saw in the Class 6 textbook of mathematics?

Answer

Write every term as a pile of tokens and push all the piles onto one table.

For 15 + (–8) + 5 + 12 the table holds 15 + 5 + 12 = 32 positive tokens and 8 negative tokens.

8 zero-pairs cancel
32 – 8 = 24 positive tokens are left
Value = 24

Why it happens: The order in which the piles are pushed onto the table does not change how many positive and negative tokens end up on it. Since the value is decided only by the leftover tokens after cancelling, every order must give the same value.

Note: The Hindi edition prints this Token-Model prompt on page 30; the English edition repeats it here on page 31.

Q3.

Manasa is adding a long list of numbers. It took her five minutes to add them all and she got the answer 11749. Then she realised that she had forgotten to include the fourth number 9055. Does she have to start all over again?

Answer

No, she does not. She only has to add the missing number to the total she already has.

Correct sum = 11749 + 9055
= 11749 + 9000 + 55
= 20749 + 55
= 20804

Why it happens: The terms of a sum may be added in any order and in any grouping (commutative and associative properties). So leaving 9055 for the very end gives exactly the same total as putting it in fourth place.

Tip: The same idea saves time in shops. If the bill is added up and one item is missed, the shopkeeper simply adds that item’s price to the total instead of re-adding the whole bill.

Q4.

Manasa is going outside to play. Her mother says, “Wear your hat and shoes!” Which one should she wear first?

Answer

It does not matter — she can wear the hat first and then the shoes, or the shoes first and then the hat. Manasa will look exactly the same either way.

hat, then shoes = shoes, then hat

But if her mother says “Wear your socks and shoes!”, the order does matter. Socks must go on before shoes.

Why it happens: Some actions commute (order does not matter) and some do not. Addition of terms commutes: 6 + (–4) = (–4) + 6. Subtraction does not: 10 – 4 is not 4 – 10 — that is exactly the “socks and shoes” situation.

Try This: List three pairs of everyday actions that commute (comb hair / drink water) and three that do not (open the bottle / pour the water).

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