NCERT Solutions Ganita Prakash (Part 1) Chapter 2 – 38Removing Brackets — Figure it Out

Book page 37 Updated on2026-09-05

Q1.

Fill in the blanks with numbers, and boxes with operation signs such that the expressions on both sides are equal. (a) 24 + (6 – 4) = 24 + 6 ☐ _____ (b) 38 + (_____ ☐ _____) = 38 + 9 – 4 (c) 24 – (6 + 4) = 24 ☐ 6 – 4 (d) 24 – 6 – 4 = 24 – 6 ☐ _____ (e) 27 – (8 + 3) = 27 ☐ 8 ☐ 3 (f) 27 – (_____ ☐ _____) = 27 – 8 + 3

Answer

A bracket with a plus before it keeps the signs inside; a bracket with a minus before it flips them.

(a) 24 + (6 – 4) = 24 + 6 – 4

(b) 38 + (9 – 4) = 38 + 9 – 4

(c) 24 – (6 + 4) = 24 – 6 – 4

(d) 24 – 6 – 4 = 24 – 6 – 4

(e) 27 – (8 + 3) = 27 – 8 – 3

(f) 27 – (8 – 3) = 27 – 8 + 3

	Left side	Right side	Both equal
(a)	24 + 2	24 + 6 – 4	26
(b)	38 + 5	38 + 9 – 4	43
(c)	24 – 10	24 – 6 – 4	14
(d)	24 – 6 – 4	24 – 6 – 4	14
(e)	27 – 11	27 – 8 – 3	16
(f)	27 – 5	27 – 8 + 3	22

Why it happens: In (c) and (e) the whole bracket is being taken away, so both numbers inside must be taken away. In (f) we are asked to put the bracket back: since 27 – 8 + 3 takes away 8 but gives back 3, the bracket must contain 8 – 3.

Q2.

Remove the brackets and write the expression having the same value. (a) 14 + (12 + 10) (b) 14 – (12 + 10) (c) 14 + (12 – 10) (d) 14 – (12 – 10) (e) –14 + (12 – 10) (f) 14 – (–12 – 10)

Answer

	Given	Brackets removed	Value
(a)	14 + (12 + 10)	14 + 12 + 10	36
(b)	14 – (12 + 10)	14 – 12 – 10	–8
(c)	14 + (12 – 10)	14 + 12 – 10	16
(d)	14 – (12 – 10)	14 – 12 + 10	12
(e)	–14 + (12 – 10)	–14 + 12 – 10	–12
(f)	14 – (–12 – 10)	14 + 12 + 10	36

(b) 14 – 12 – 10 = 2 – 10 = –8
(d) 14 – 12 + 10 = 2 + 10 = 12
(f) the minus flips both –12 and –10 to +12 and +10 → 14 + 12 + 10 = 36

Why it happens: A bracket after a ‘+’ changes nothing. A bracket after a ‘–’ flips the sign of every term inside. So in (f), taking away (–12 – 10) means taking away –22, which is the same as adding 22.

Did you know? (a) and (f) have the same value 36, even though (f) looks full of minus signs.

Q3.

Find the values of the following expressions. For each pair, first try to guess whether they have the same value. When are the two expressions equal? (a) (6 + 10) – 2 and 6 + (10 – 2) (b) 16 – (8 – 3) and (16 – 8) – 3 (c) 27 – (18 + 4) and 27 + (–18 – 4)

Answer

(a) (6 + 10) – 2 = 16 – 2 = 14
     6 + (10 – 2) = 6 + 8 = 14 → equal
     Both are 6 + 10 + (–2).

(b) 16 – (8 – 3) = 16 – 5 = 11
     (16 – 8) – 3 = 8 – 3 = 5 → not equal
     First is 16 – 8 + 3, second is 16 – 8 – 3.

(c) 27 – (18 + 4) = 27 – 22 = 5
     27 + (–18 – 4) = 27 – 22 = 5 → equal
     Both are 27 + (–18) + (–4).

When are the two expressions equal? They are equal exactly when, after removing the brackets, both expressions become the same list of terms with the same signs.

Why it happens: In (b) the bracket sits after a minus sign, so 16 – (8 – 3) becomes 16 – 8 + 3 while (16 – 8) – 3 stays 16 – 8 – 3. The sign of 3 is different, so the values differ by 6.

Q4.

In each of the sets of expressions below, identify those that have the same value. Do not evaluate them, but rather use your understanding of terms. (a) 319 + 537, 319 – 537, –537 + 319, 537 – 319 (b) 87 + 46 – 109, 87 + 46 – 109, 87 + 46 – 109, 87 – 46 + 109, 87 – (46 + 109), (87 – 46) + 109

Answer

(a) Write each as a sum of terms:

Expression	Terms
319 + 537	319, 537
319 – 537	319, –537
–537 + 319	–537, 319
537 – 319	537, –319

319 – 537 and –537 + 319 have the same value — the same two terms, only swapped.

(b) Again compare the terms:

Expression	Terms	Group
87 + 46 – 109 (three times)	87, 46, –109	Group 1
87 – 46 + 109	87, –46, 109	Group 2
87 – (46 + 109)	87, –46, –109	on its own
(87 – 46) + 109	87, –46, 109	Group 2

So the three copies of 87 + 46 – 109 are equal to one another, and 87 – 46 + 109 = (87 – 46) + 109. The odd one out is 87 – (46 + 109).

Why it happens: Two expressions built only from additions are equal when they carry the same terms with the same signs, no matter what order they appear in. Removing the bracket in 87 – (46 + 109) flips 109 to –109, which puts it in a group of its own.

Check it yourself: Group 1 = 24, Group 2 = 150, and 87 – (46 + 109) = –68.

Q5.

Add brackets at appropriate places in the expressions such that they lead to the values indicated. (a) 34 – 9 + 12 = 13 (b) 56 – 14 – 8 = 34 (c) –22 – 12 + 10 + 22 = –22

Answer

(a) 34 – (9 + 12) = 34 – 21 = 13
(without the bracket 34 – 9 + 12 would be 37)

(b) (56 – 14) – 8 = 42 – 8 = 34

(c) –22 – (12 + 10) + 22 = –22 – 22 + 22 = –22

Why it happens: In (a) the bracket makes 12 be taken away instead of added, a change of 24 (37 → 13). In (c) the bracket after the minus flips +10 into –10, so the middle part becomes –22, which cancels the last +22.

Tip: In (b) the value is already 34 without any bracket. The bracket only makes the intended order explicit — that is allowed.

Q6.

Using only reasoning of how terms change their values, fill the blanks to make the expressions on either side of the equality (=) equal. (a) 423 + ______ = 419 + ______ (b) 207 – 68 = 210 – ______

Answer

(a) 423 + 419 = 419 + 423
     (commutative property — the same two terms, swapped)

     Another answer: 423 + 6 = 419 + 10
     (419 is 4 less than 423, so the second term must be 4 more)

(b) 207 – 68 = 210 – 71
     210 is 3 more than 207, so we must subtract 3 more: 68 + 3 = 71
     Check: 207 – 68 = 139 and 210 – 71 = 139 ✓

Why it happens: If the first term goes up by 3, the amount taken away must also go up by 3 for the value to stay the same. Any change on one side must be balanced on the other.

Q7.

Using the numbers 2, 3 and 5, and the operators ‘+’ and ‘–’, and brackets, as necessary, generate expressions to give as many different values as possible. For example, 2 – 3 + 5 = 4 and 3 – (5 – 2) = 0.

Answer

Use each of 2, 3 and 5 once. Here are all the different values that can be reached.

Value	Expressions
10	2 + 3 + 5; 5 + 3 + 2
6	3 – 2 + 5; 5 – 2 + 3; 5 + 3 – 2
4	2 – 3 + 5; 5 + 2 – 3; 5 – (3 – 2)
0	2 + 3 – 5; 3 – (5 – 2); 5 – 3 – 2; 5 – (3 + 2)
–4	3 – 2 – 5; 3 – (2 + 5)
–6	2 – 3 – 5; 2 – (3 + 5)

So we get six different values: –6, –4, 0, 4, 6 and 10.

Why it happens: With only + and –, the value is 2, 3 and 5 each taken with a plus or a minus sign. Since the first number written is always added, the six sign patterns give only these six sums. Brackets do not add new values here — they only give new ways of writing the same six.

Try This: Repeat with 1, 4 and 6 and see how many different values you get.

Q8.

Whenever Jasoda has to subtract 9 from a number, she subtracts 10 and adds 1 to it. For example, 36 – 9 = 26 + 1. (a) Do you think she always gets the correct answer? Why? (b) Can you think of other similar strategies? Give some examples.

Answer

(a) Yes, she always gets the correct answer.

9 = 10 – 1
So, any number – 9 = number – (10 – 1)
= number – 10 + 1 (bracket after a minus flips the signs)

36 – 9 = 36 – 10 + 1 = 26 + 1 = 27 ✓
84 – 9 = 84 – 10 + 1 = 74 + 1 = 75 ✓
123 – 9 = 123 – 10 + 1 = 113 + 1 = 114 ✓

(b) Other strategies of the same kind:

To do this	Do this instead	Example
subtract 8	subtract 10, add 2	55 – 8 = 55 – 10 + 2 = 47
subtract 99	subtract 100, add 1	432 – 99 = 432 – 100 + 1 = 333
subtract 19	subtract 20, add 1	76 – 19 = 76 – 20 + 1 = 57
add 9	add 10, subtract 1	48 + 9 = 48 + 10 – 1 = 57
multiply by 9	multiply by 10, subtract the number	23 × 9 = 230 – 23 = 207

Why it happens: Round numbers like 10 and 100 are easy to subtract. Writing the awkward number as a round number minus a small correction turns one hard step into two easy ones.

Q9.

Consider the two expressions: a) 73 – 14 + 1, b) 73 – 14 – 1. For each of these expressions, identify the expressions from the following collection that are equal to it. (a) 73 – (14 + 1) (b) 73 – (14 – 1) (c) 73 + (–14 + 1) (d) 73 + (–14 – 1)

Answer

Remove every bracket first, then compare the terms.

Expression	Without brackets	Terms	Value
(a) 73 – (14 + 1)	73 – 14 – 1	73, –14, –1	58
(b) 73 – (14 – 1)	73 – 14 + 1	73, –14, 1	60
(c) 73 + (–14 + 1)	73 – 14 + 1	73, –14, 1	60
(d) 73 + (–14 – 1)	73 – 14 – 1	73, –14, –1	58

73 – 14 + 1 = 60 is equal to (b) and (c)
73 – 14 – 1 = 58 is equal to (a) and (d)

Why it happens: Everything turns on the sign of the 1. A bracket after a minus flips it — that is why 73 – (14 – 1) has +1 inside. A bracket after a plus leaves it alone.

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