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NCERT Solutions Ganita Prakash (Part 1) Chapter 8 –198Chapter Exercises — Figure it Out

Book page 196 Updated on2026-09-05

Q1.
Evaluate the following: 3 ÷ 7⁄9;   14⁄4 ÷ 2;   2⁄3 ÷ 2⁄3;   14⁄6 ÷ 7⁄3;   4⁄3 ÷ 3⁄4;   7⁄4 ÷ 1⁄7;   8⁄2 ÷ 4⁄15;   1⁄5 ÷ 1⁄9;   1⁄6 ÷ 11⁄12;   3 2⁄3 ÷ 1 3⁄8
Answer

Turn every divisor upside down and multiply.

DivisionMultiply by the reciprocalAnswer
3 ÷ 7/93 × 9/727/7 = 3 6/7
14/4 ÷ 214/4 × 1/214/8 = 7/4 = 1 3/4
2/3 ÷ 2/32/3 × 3/21
14/6 ÷ 7/314/6 × 3/742/42 = 1
4/3 ÷ 3/44/3 × 4/316/9 = 1 7/9
7/4 ÷ 1/77/4 × 749/4 = 12 1/4
8/2 ÷ 4/154 × 15/415
1/5 ÷ 1/91/5 × 99/5 = 1 4/5
1/6 ÷ 11/121/6 × 12/1112/66 = 2/11
3 2/3 ÷ 1 3/811/3 × 8/118/3 = 2 2/3
Tip: change mixed fractions first — 3 2⁄3 = 11⁄3 and 1 3⁄8 = 11⁄8. The 11s then cancel beautifully.
Q2.
For each of the questions below, choose the expression that describes the solution. Then simplify it. (a) Maria bought 8 m of lace to decorate the bags she made for school. She used 1⁄4 m for each bag and finished the lace. How many bags did she decorate? (i) 8 × 1⁄4 (ii) 1⁄8 × 1⁄4 (iii) 8 ÷ 1⁄4 (iv) 1⁄4 ÷ 8.   (b) 1⁄2 meter of ribbon is used to make 8 badges. What is the length of the ribbon used for each badge? (i) 8 × 1⁄2 (ii) 1⁄2 ÷ 1⁄8 (iii) 8 ÷ 1⁄2 (iv) 1⁄2 ÷ 8.   (c) A baker needs 1⁄6 kg of flour to make one loaf of bread. He has 5 kg of flour. How many loaves of bread can he make? (i) 5 × 1⁄6 (ii) 1⁄6 ÷ 5 (iii) 5 ÷ 1⁄6 (iv) 5 × 6
Answer

(a) Option (iii) — how many quarter-metre pieces are in 8 m?

8 ÷ 1/4 = 8 × 4 = 32 bags

(b) Option (iv) — half a metre is shared among 8 badges.

1/2 ÷ 8 = 1/2 × 1/8 = 1/16 m (6.25 cm each)

(c) Option (iii) — how many sixth-kilogram scoops are in 5 kg?

5 ÷ 1/6 = 5 × 6 = 30 loaves
How to choose: ask what kind of question it is. “How many small pieces fit inside?” is division by the piece. “One whole shared among many?” is division by the number of shares. In (c), option (iv) gives the right number by luck, but 5 × 6 is not the expression that describes the situation.
Q3.
If 1⁄4 kg of flour is used to make 12 rotis, how much flour is used to make 6 rotis?
Answer

Six rotis are half of twelve rotis.

Flour for 1 roti = 1/4 ÷ 12 = 1/48 kg
Flour for 6 rotis = 6 × 1/48
= 6/48
= 1/8 kg
Shortcut: half the rotis need half the flour — half of 1⁄4 kg is 1⁄8 kg (125 g).
Q4.
Pāṭīgaṇita, a book written by Sridharacharya in the 9th century CE, mentions this problem: “Friend, after thinking, what sum will be obtained by adding together 1 ÷ 1⁄6, 1 ÷ 1⁄10, 1 ÷ 1⁄13, 1 ÷ 1⁄9, and 1 ÷ 1⁄2”. What should the friend say?
Answer

Dividing 1 by a unit fraction just gives back its denominator.

1 ÷ 1/6 = 6
1 ÷ 1/10 = 10
1 ÷ 1/13 = 13
1 ÷ 1/9 = 9
1 ÷ 1/2 = 2
Sum = 6 + 10 + 13 + 9 + 2 = 40

The friend should say 40.

Why it happens: 1 ÷ 1⁄n asks “how many 1⁄n pieces make one whole?” — and the answer is always n.
Q5.
Mira is reading a novel that has 400 pages. She read 1⁄5 of the pages yesterday and 3⁄10 of the pages today. How many more pages does she need to read to finish the novel?
Answer

Find the pages read on each day, then subtract from 400.

Yesterday = 1/5 × 400 = 80 pages
Today = 3/10 × 400 = 120 pages
Read so far = 80 + 120 = 200 pages
Left = 400 − 200 = 200 pages
Another way: 1/5 + 3/10 = 2/10 + 3/10 = 5/10 = 1/2. She has read exactly half the book, so half of 400 = 200 pages remain.
Q6.
A car runs 16 km using 1 litre of petrol. How far will it go using 2 3⁄4 litres of petrol?
Answer

Multiply the mileage by the number of litres.

2 3/4 litres = 11/4 litres
Distance = 11/4 × 16
= 11 × 4
= 44 km
Why it works: 16 km per litre, so 2 litres give 32 km and the extra 3⁄4 litre gives 3⁄4 × 16 = 12 km. Together 32 + 12 = 44 km.
Q7.
Amritpal decides on a destination for his vacation. If he takes a train, it will take him 5 1⁄6 hours to get there. If he takes a plane, it will take him 1⁄2 hour. How many hours does the plane save?
Answer

The time saved is the difference of the two times.

5 1/6 = 31/6 hours
Saving = 31/6 − 1/2
= 31/6 − 3/6
= 28/6
= 14/3 hours = 4 2/3 hours

The plane saves 4 2⁄3 hours, that is 4 hours 40 minutes.

Check it yourself: 1⁄2 + 4 2⁄3 = 3⁄6 + 28⁄6 = 31⁄6 = 5 1⁄6 ✔
Q8.
Mariam’s grandmother baked a cake. Mariam and her cousins finished 4⁄5 of the cake. The remaining cake was shared equally by Mariam’s three friends. How much of the cake did each friend get?
Answer

Find what is left, then split it three ways.

Cake left = 1 − 4/5 = 1/5
Each friend = 1/5 ÷ 3
= 1/5 × 1/3
= 1/15 of the cake
Why it happens: one-fifth of the cake cut into 3 equal pieces gives fifteenths, because 5 × 3 = 15 such pieces would make the whole cake.
Q9.
Choose the option(s) describing the product of (565⁄465 × 707⁄676): (a) > 565⁄465 (b) < 565⁄465 (c) > 707⁄676 (d) < 707⁄676 (e) > 1 (f) < 1
Answer

You do not have to multiply — just check where each fraction sits compared with 1.

565 > 465, so 565/465 > 1
707 > 676, so 707/676 > 1

Both numbers are greater than 1. So the product is greater than each of them, and greater than 1.

Correct options: (a), (c) and (e).

Check it yourself: 565⁄465 ≈ 1.22 and 707⁄676 ≈ 1.05, so the product ≈ 1.27 — bigger than 1.22, bigger than 1.05, bigger than 1 ✔
Q10.
What fraction of the whole square is shaded?
Answer

The shading lies entirely inside the bottom-right quarter of the square.

the “Y” splits the bottom-right quarter
Inside the bottom-right quarter, the shaded part is a triangle plus a square — 3 of its 8 equal halves.
Take that quarter as 1 unit.
The shading fills the left half of the quarter, except one small triangle at its top.
Left half of the quarter = 1/2
Triangle cut away = 1/2 × 1/2 × 1/2 = 1/8
Shaded part of the quarter = 1/2 − 1/8 = 3/8

Shaded part of the whole square = 3/8 × 1/4
= 3/32
How to read the picture: the two slanting lines start at the top corners of the quarter and meet at its centre; from there a straight line drops to the bottom edge. That line is exactly halfway across, so the shaded piece is the left half of the quarter minus the small triangle above the slanting line.
Q11.
A colony of ants set out in search of food. As they search, they keep splitting equally at each point (as shown in the Fig. 8.7) and reach two food sources, one near a mango tree and another near a sugarcane field. What fraction of the original group reached each food source?
Answer

Start with the whole colony as 1 and halve it at every splitting point.

SplitGoes to mango treeCarries on
1st point1/21/2
2nd point1/41/4
3rd point1/16 + 1/16 (that branch splits again)1/8 → splits into 1/16 and 1/16
4th point1/321/32 → sugarcane field
Mango tree = 1/2 + 1/4 + 1/16 + 1/16 + 1/32
= 16/32 + 8/32 + 2/32 + 2/32 + 1/32
= 29/32

Sugarcane field = 1/16 + 1/32
= 2/32 + 1/32
= 3/32
Check it yourself: 29⁄32 + 3⁄32 = 32⁄32 = 1 — the whole colony is accounted for ✔ Almost all the ants end up at the mango tree.
Q12.
What is 1 − 1⁄2?   (1 − 1⁄2) × (1 − 1⁄3)?   (1 − 1⁄2) × (1 − 1⁄3) × (1 − 1⁄4) × (1 − 1⁄5)?   (1 − 1⁄2) × (1 − 1⁄3) × (1 − 1⁄4) × (1 − 1⁄5) × (1 − 1⁄6) × (1 − 1⁄7) × (1 − 1⁄8) × (1 − 1⁄9) × (1 − 1⁄10)? Make a general statement and explain.
Answer

Turn each bracket into a single fraction first.

1 − 1/2 = 1/2

(1 − 1/2) × (1 − 1/3) = 1/2 × 2/3 = 1/3

1/2 × 2/3 × 3/4 × 4/5 = 1/5

1/2 × 2/3 × 3/4 × 4/5 × 5/6 × 6/7 × 7/8 × 8/9 × 9/10 = 1/10

General statement:

(1 − 1/2) × (1 − 1/3) × (1 − 1/4) × … × (1 − 1/n) = 1/n
Why it happens: 1 − 1⁄k = (k − 1)⁄k. So the chain reads 1⁄2 × 2⁄3 × 3⁄4 × … × (n − 1)⁄n. Every denominator cancels with the next numerator — a telescoping product — leaving only the first numerator 1 and the last denominator n.
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