NCERT Solutions Ganita Prakash (Part 1) Chapter 2 – 42The Distributive Property — Figure it Out

Book page 41 Updated on2026-09-05

Q1.

Fill in the blanks with numbers, and boxes by signs, so that the expressions on both sides are equal. (a) 3 × (6 + 7) = 3 × 6 + 3 × 7 (b) (8 + 3) × 4 = 8 × 4 + 3 × 4 (c) 3 × (5 + 8) = 3 × 5 ☐ 3 × ____ (d) (9 + 2) × 4 = 9 × 4 ☐ 2 × ____ (e) 3 × (____ + 4) = 3 ____ + ____ (f) (____ + 6) × 4 = 13 × 4 + ____ (g) 3 × (____ + ____) = 3 × 5 + 3 × 2 (h) (____ + ____) × ____ = 2 × 4 + 3 × 4 (i) 5 × (9 – 2) = 5 × 9 – 5 × ____ (j) (5 – 2) × 7 = 5 × 7 – 2 × ____ (k) 5 × (8 – 3) = 5 × 8 ☐ 5 × ____ (l) (8 – 3) × 7 = 8 × 7 ☐ 3 × 7 (m) 5 × (12 – ____) = ____ ☐ 5 × ____ (n) (15 – ____) × 7 = ____ ☐ 6 × 7 (o) 5 × (____ – ____) = 5 × 9 – 5 × 4 (p) (____ – ____) × ____ = 17 × 7 – 9 × 7

Answer

Multiply the outside number into every term inside the bracket, keeping each sign.

(a) 3 × (6 + 7) = 3 × 6 + 3 × 7 (given)
(b) (8 + 3) × 4 = 8 × 4 + 3 × 4 (given)
(c) 3 × (5 + 8) = 3 × 5 + 3 × 8
(d) (9 + 2) × 4 = 9 × 4 + 2 × 4
(e) 3 × (10 + 4) = 3 × 10 + 3 × 4
(f) (13 + 6) × 4 = 13 × 4 + 6 × 4
(g) 3 × (5 + 2) = 3 × 5 + 3 × 2
(h) (2 + 3) × 4 = 2 × 4 + 3 × 4
(i) 5 × (9 – 2) = 5 × 9 – 5 × 2
(j) (5 – 2) × 7 = 5 × 7 – 2 × 7
(k) 5 × (8 – 3) = 5 × 8 – 5 × 3
(l) (8 – 3) × 7 = 8 × 7 – 3 × 7
(m) 5 × (12 – 3) = 5 × 12 – 5 × 3
(n) (15 – 6) × 7 = 15 × 7 – 6 × 7
(o) 5 × (9 – 4) = 5 × 9 – 5 × 4
(p) (17 – 9) × 7 = 17 × 7 – 9 × 7

Why it happens: The distributive property says a × (b + c) = a × b + a × c and a × (b – c) = a × b – a × c. In (e) and (m) any number may be used in the first blank, as long as the same number is repeated on the right — for example 3 × (7 + 4) = 3 × 7 + 3 × 4 also works.

Check it yourself: (m) 5 × 9 = 45 and 60 – 15 = 45 ✓ ; (n) 9 × 7 = 63 and 105 – 42 = 63 ✓

Q2.

In the boxes below, fill ‘<’, ‘>’ or ‘=’ after analysing the expressions on the LHS and RHS. Use reasoning and understanding of terms and brackets to figure this out and not by evaluating the expressions. (a) (8 – 3) × 29 ☐ (3 – 8) × 29 (b) 15 + 9 × 18 ☐ (15 + 9) × 18 (c) 23 × (17 – 9) ☐ 23 × 17 + 23 × 9 (d) (34 – 28) × 42 ☐ 34 × 42 – 28 × 42

Answer

(a) (8 – 3) × 29 > (3 – 8) × 29
     8 – 3 is positive (5) while 3 – 8 is negative (–5); the same multiplier 29 keeps the order.

(b) 15 + 9 × 18 < (15 + 9) × 18
     LHS multiplies only the 9 by 18; RHS multiplies both 15 and 9 by 18, adding 15 × 17 extra.

(c) 23 × (17 – 9) < 23 × 17 + 23 × 9
     LHS = 23 × 17 – 23 × 9, so it subtracts 23 × 9 while the RHS adds it.

(d) (34 – 28) × 42 = 34 × 42 – 28 × 42
     This is exactly the distributive property.

Why it happens: No multiplication is actually needed. In (c) both sides share the piece 23 × 17; one side then takes away 23 × 9 and the other adds it, so the side that adds must be bigger.

Check it yourself: (a) 145 > –145 ✓ (b) 177 < 432 ✓ (c) 184 < 598 ✓ (d) 252 = 252 ✓

Q3.

Here is one way to make 14: _2_ × ( _1_ + _6_ ) = 14. Are there other ways of getting 14? Fill them out below: (a) ____ × (____ + ____) = 14 (b) ____ × (____ + ____) = 14 (c) ____ × (____ + ____) = 14 (d) ____ × (____ + ____) = 14

Answer

Split 14 as a product first, then split one factor as a sum.

14 = 2 × 7 = 7 × 2 = 1 × 14 = 14 × 1

	Expression	Check
(a)	2 × (5 + 2) = 14	2 × 7 = 14
(b)	2 × (3 + 4) = 14	2 × 7 = 14
(c)	7 × (1 + 1) = 14	7 × 2 = 14
(d)	14 × (0 + 1) = 14	14 × 1 = 14

More answers: 1 × (7 + 7), 2 × (6 + 1), 7 × (0 + 2).

Why it happens: The bracket must add up to a number that divides 14 exactly — that is 1, 2, 7 or 14. Once the bracket total is fixed, the two numbers inside can be split in any way you like.

Q4.

Find out the sum of the numbers given in each picture below in at least two different ways. Describe how you solved it through expressions.

Answer

Picture (I) — a 3 × 3 arrangement: five yellow squares showing 4 and four blue circles showing 8, placed like the dots on a die.

4	8	4
8	4	8
4	8	4

Way 1 (by colour): 5 × 4 + 4 × 8
= 20 + 32 = 52

Way 2 (row by row): (4 + 8 + 4) + (8 + 4 + 8) + (4 + 8 + 4)
= 16 + 20 + 16 = 52

Way 3: 2 × (4 + 8 + 4) + (8 + 4 + 8)
= 2 × 16 + 20 = 32 + 20 = 52

Picture (II) — a 4 × 4 arrangement of circles: eight blue circles showing 5 and eight red circles showing 6.

5	6	6	5
6	5	5	6
6	5	5	6
5	6	6	5

Way 1 (by colour): 8 × 5 + 8 × 6
= 40 + 48 = 88

Way 2 (distributive): 8 × (5 + 6)
= 8 × 11 = 88

Way 3 (row by row): each row adds to 22, and there are 4 rows
4 × 22 = 88

Why it happens: Way 1 and Way 2 of Picture (II) are the two sides of the distributive property: 8 × 5 + 8 × 6 = 8 × (5 + 6). Grouping by colour or by row simply counts the same numbers in a different order, so the total cannot change.

Tip: Look for a pattern before adding. In Picture (I) the four corners and the centre are all 4s, and the four edge-middles are all 8s — that gives 5 × 4 + 4 × 8 straight away.

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