NCERT Solutions Class 7 Maths Chapter 8 Working with Fractions

F R E E S T U D Y M AT E R I A L F O R E V E R Y S T U D E N T

C L A S S 7 · M AT H S

NCERT Solutions

Chapter 8: Working with
Fractions

NCERT Textbook — Ganita Prakash

BOOK PAGES SECTIONS QUESTIONS MEDIUM

173 – 199 16 62 English

Solutions, notes, sample papers & more at 47 pages

Class 7 Maths Chapter 8 Working with Fractions AglaSem · NCERT Solutions

CLASS 7 · MATHS · GANITA PRAKASH

NCERT Solutions — Chapter 8: Working with Fractions
Chapter 8 of Ganita Prakash Part I takes fractions beyond adding and subtracting. Using a walking tortoise, a
unit square and the area of a rectangle, the chapter builds multiplication and division of fractions — and
shows that both rules were written down in general form by Brahmagupta in 628 CE.

TEXTBOOK BOOK PAGES

Ganita Prakash (Class 7) 173 – 199

SECTIONS QUESTIONS

16 62

MEDIUM

English

In-text Questions — Pages 173–175
Section 8.1 Multiplication of Fractions

Q1 Aaron walks 3 kilometres in 1 hour. How far can he walk in 5 hours?

Walking for 5 hours means covering the 1-hour distance 5 times.

Distance covered in 1 hour = 3 km

Distance covered in 5 hours = 5 × 3 km

= 3 + 3 + 3 + 3 + 3 km

= 15 km

Why it happens: multiplication is just repeated addition. Each new hour adds
another 3 km to the total, so 5 hours add 3 km five times.

Q2 Aaron’s pet tortoise walks at a much slower pace. It can walk only 1⁄4 kilometre in 1
hour. How far can it walk in 3 hours?

The speed is a fraction now, but nothing else changes — still multiply.

Page 1 of 47

Class 7 Maths Chapter 8 Working with Fractions AglaSem · NCERT Solutions

Distance in 1 hour = 1/4 km

Distance in 3 hours = 3 × 1/4 km

= 1/4 + 1/4 + 1/4 km
= 3/4 km

The tortoise can walk 3⁄4 km in 3 hours.

Tip: three quarter-kilometres make three-quarters of a kilometre — 750 metres. Still
less than 1 km, which fits a slow tortoise.

Q3 We saw that Aaron can walk 3 kilometres in 1 hour. How far can he walk in 1⁄5
hours?

One-fifth of an hour covers one-fifth of the 1-hour distance.

Distance in 1 hour = 3 km

Distance in 1/5 hour = 3 km divided into 5 equal parts

=3÷5

= 3/5 km

So 1⁄5 × 3 = 3⁄5.

Why it happens: five stretches of 1⁄5 hour make one full hour. So five equal pieces of
the distance must make 3 km, and each piece is 3⁄5 km.

Q4 How far can Aaron walk in 2⁄5 hours?

Work in two steps — first one-fifth of an hour, then double it.

Page 2 of 47

Class 7 Maths Chapter 8 Working with Fractions AglaSem · NCERT Solutions

Step 1: distance in 1/5 hour = 3 ÷ 5 = 3/5 km

Step 2: 2/5 hour is twice 1/5 hour

Distance = 2 × 3/5
= 6/5 km = 1 1/5 km

So 2⁄5 × 3 = 6⁄5.

Tip: this is the whole method in a nutshell — divide by the denominator of the
multiplier, then multiply by its numerator. Every fraction multiplication in this
chapter follows these two steps.

In-text Questions — Page 176
Section 8.1 Worked Examples

Q1 Example 1: A farmer had 5 grandchildren. She distributed 2⁄3 acre of land to each of
her grandchildren. How much land in all did she give to her grandchildren?

Five equal shares of 2⁄3 acre each.

5 × 2/3 = 2/3 + 2/3 + 2/3 + 2/3 + 2/3

= 10/3

= 3 1/3 acres

Why it happens: ten one-third pieces are handed out in all. Three thirds make one
acre, so ten thirds make 3 whole acres and one third left over.

Q2 Example 2: 1 hour of internet time costs ₹8. How much will 1 1⁄4 hours of internet
time cost?

First change the mixed fraction into an improper fraction.

Page 3 of 47

as e
Class 7 Maths Chapter 8 Working with Fractions
a g l AglaSem · NCERT Solutions

co m
e m.
1 1/4 hours = 5/4 hours
m l as
Cost = 5/4 × 8
m .co a g
l a se
g
= (5 × 8) ÷ 4
a= 5 × 2

co m
. ag
= ₹10
em
g l as
a
Tip: divide first (8 ÷ 4 = 2), then multiply (5 × 2 = 10). Smaller numbers, fewer
mistakes.
co m
em.
m l as
m .co a g
ase it Out — Pages 176–177
agl
Figure
Section 8.1 Multiplication of Fractions

m a s
Tenzin drinks 1⁄
e
. c o
m day. How many glasses of milk does he drink in agl
s
Q1 2 glass of milk every
aof milk did he drink in the month of January?
agl
a week? How many glasses

co m
m .
Multiply the daily amount by the number of days.
m as e
.co a g l
a s eInma week (7 days) = 7 × 1/2 = 7/2 = 3 1/2 glasses
agl
m
January has 31 days

a se
In January = 31 × 1/2 = 31/2 = 15 1/2 glasses
. com a g l
m
ase
agl
Why it happens: two half-glasses make one full glass. Seven halves are three full
glasses and one half left; thirty-one halves are fifteen full glasses and one half left.
co m
m .
m as e
.co a g l
a s emQ2
gl
A team of workers can make 1 km of a water canal in 8 days. So, in one day, the
a team can make ___ km of the water canal. If they work 5 days a week, they can
c
m .
e
make ___ km of the water canal in a week.
m a s
m . co agl
l a se
ag equally over 8 days.
One kilometre of work is spread

co m
m .
m ase
.co


a g l Page 4 of 47

Class 7 Maths Chapter 8 Working with Fractions AglaSem · NCERT Solutions

Work in 1 day = 1 ÷ 8 = 1/8 km

Work in 5 days = 5 × 1/8 = 5/8 km

DAYS WORKED CANAL MADE

1 1/8 km

5 (one week) 5/8 km

8 8/8 = 1 km

Check it yourself: 5/8 km = 625 m — a little more than half a kilometre in a 5-day
week.

Q3 Manju and two of her neighbours buy 5 litres of oil every week and share it equally
among the 3 families. How much oil does each family get in a week? How much oil
will one family get in 4 weeks?

Five litres shared equally by 3 families.

Oil per family in 1 week = 5 ÷ 3 = 5/3 litres = 1 2/3 litres

In 4 weeks = 4 × 5/3 = 20/3 = 6 2/3 litres

Why it happens: cut every litre into 3 equal parts. There are 5 × 3 = 15 one-third
parts, and 15 ÷ 3 = 5 parts for each family — that is 5⁄3 litres.

Q4 Safia saw the Moon setting on Monday at 10 pm. Her mother, who is a scientist, told
her that every day the Moon sets 5⁄6 hour later than the previous day. How many
hours after 10 pm will the moon set on Thursday?

Monday to Thursday is 3 days, so the delay builds up three times.

Page 5 of 47

Class 7 Maths Chapter 8 Working with Fractions AglaSem · NCERT Solutions

Delay = 3 × 5/6

= 15/6

= 5/2
= 2 1/2 hours after 10 pm

So the Moon sets at about 12:30 am (past midnight).

Tip: Monday → Tuesday is 1 delay, Tuesday → Wednesday is 2, Wednesday →
Thursday is 3. Count the gaps between days, not the days.

Q5 Multiply and then convert it into a mixed fraction: (a) 7 × 3⁄5 (b) 4 × 1⁄3 (c) 9⁄7 × 6 (d)
13⁄
11 × 6

Multiply the whole number into the numerator, then split off the whole part.

QUESTION PRODUCT MIXED FRACTION

(a) 7 × 3/5 21/5 4 1/5

(b) 4 × 1/3 4/3 1 1/3

(c) 9/7 × 6 54/7 7 5/7

(d) 13/11 × 6 78/11 7 1/11

21 ÷ 5 = 4 remainder 1 → 4 1/5

54 ÷ 7 = 7 remainder 5 → 7 5/7

78 ÷ 11 = 7 remainder 1 → 7 1/11

Why it works: the quotient tells you how many wholes are hidden inside the
improper fraction and the remainder is the fifths (or sevenths, or elevenths) that are
left over.

Page 6 of 47

Class 7 Maths Chapter 8 Working with Fractions AglaSem · NCERT Solutions

In-text Questions — Pages 177–179
Multiplying Two Fractions

MATH TALK

Q1 We know, that Aaron’s pet tortoise can walk only 1⁄4 km in 1 hour. How far can it
walk in half an hour?

Half an hour covers half of the one-hour distance.

Distance = 1/2 × 1/4 km

= 1/4 divided into 2 equal parts

= 1/8 km

1/8

whole = 4 rows × 2 columns =
8 equal parts

Page 7 of 47

Class 7 Maths Chapter 8 Working with Fractions AglaSem · NCERT Solutions

The unit square is cut into 4 rows and 2 columns. One-quarter (top two cells) is halved, and the dark
cell is 1 of the 8 equal parts.

Why it happens: cutting the whole into 4 rows and then into 2 columns makes 4 × 2
= 8 equal parts, so each part is 1⁄8 of the whole.

Q2 Now we divide this 1⁄4 into 2 equal parts. What do we get? What fraction of the
whole is shaded?

The whole is now in 8 equal parts and one of them is shaded.

Whole ÷ 8 equal parts

Shaded = 1 part

= 1/8 of the whole

So 1⁄2 × 1⁄4 = 1⁄8, and the tortoise walks 1⁄8 km in half an hour.

Tip: halving a quarter always gives an eighth — the number of parts doubles, so the
size of each part halves.

Q3 If the tortoise walks faster and it can cover 2⁄5 km in 1 hour, how far will it walk in 3⁄4
of an hour?

Same two steps: divide by 4, then multiply by 3.

Page 8 of 47

as e
Class 7 Maths Chapter 8 Working with Fractions
a g l AglaSem · NCERT Solutions

co m
e m.
(i) Distance in 1/4 hour = 2/5 ÷ 4
m l as
.co
The whole is cut into 5 rows and 4 columns = 20 equal parts
m a g
l a se
g
Parts shaded = 2, so distance = 2/20
a

co m
. ag
(ii) Distance in 3/4 hour = 3 × 2/20
e m
= 6/20
g l as
= 3/10 km a

co m
em.
as
So 3⁄ × 2⁄ = 6⁄ = 3⁄ 10.
m l
4 5 20

.co a g
a s emit happens: the 5 rows come from the denominator of the multiplicand and the
gl
Why
a 4 columns from the denominator of the multiplier. That is exactly why the

s
denominators get multiplied.
m a
m .co agl
l a se
a g
Using this understanding, multiply 5⁄4 × 3⁄2.
Q4

co m
m .

m as e
.co
Draw 3⁄2 as one whole and a half, then cut it into 4 equal parts.
a g l
se m
g l a
a
m
Whole cut into 2 rows and 4 columns = 8 equal parts

a se
. com g l
3/2 ÷ 4 = 3 of those parts = 3/8
m a
ase
agl
Now multiply by 5:

5/4 × 3/2 = (5 × 3)/8

= 15/8 = 1 7/8
co m
m .
o m l a se
g
.cCheck it yourself: Brahmagupta's rule gives the sameaanswer
se m straight away — (5 ×

g l a
a
3)/(4 × 2) = 15/8. Both numbers are greater than 1, so the product is greater than

.c
m
both. ✔

m a s e
m . co agl
l a se
g 180
In-text Questions —aPage

co m
m .
m ase
.co


a g l Page 9 of 47

Class 7 Maths Chapter 8 Working with Fractions AglaSem · NCERT Solutions

Connection between the Area of a Rectangle and Fraction Multiplication

Q1 In the Fig. 8.3, what is the length and breadth of the shaded rectangle?

1
8

Fig. 8.3, page 180 — the shaded rectangle inside the unit square.

The big square is a unit square, so each side is 1 unit long.

The square is cut into 2 columns → each column is 1/2 unit wide

The square is cut into 4 rows → each row is 1/4 unit tall

Length = 1/2 unit, Breadth = 1/4 unit

Why it happens: reading a length off a grid means counting how many equal parts
the side was cut into. Two equal parts of 1 unit make halves; four equal parts make
quarters.

Page 10 of 47

Class 7 Maths Chapter 8 Working with Fractions AglaSem · NCERT Solutions

Q2 What is the area of this rectangle? Do you see any relation between the area and
the product of length and breadth?

Eight of these rectangles tile the unit square.

8 rectangles = 1 square unit

Area of one rectangle = 1 ÷ 8 = 1/8 square unit

Length × breadth = 1/2 × 1/4 = 1/8

They are equal — the area of a rectangle with fractional sides is the product of its sides.

Tip: this gives you a picture for every fraction multiplication. To find a⁄b × c⁄d, draw a
rectangle with those sides inside a unit square and read off its area.

Figure it Out — Pages 180–181
Multiplying Fractional Units

Q1 Find the following products. Use a unit square as a whole for representing the
fractions: (a) 1⁄3 × 1⁄5 (b) 1⁄4 × 1⁄3 (c) 1⁄5 × 1⁄2 (d) 1⁄6 × 1⁄5

Cut the unit square into rows (first denominator) and columns (second denominator), then
shade one cell.

QUESTION ROWS × COLUMNS PRODUCT

(a) 1/3 × 1/5 3 × 5 = 15 parts 1/15

(b) 1/4 × 1/3 4 × 3 = 12 parts 1/12

(c) 1/5 × 1/2 5 × 2 = 10 parts 1/10

(d) 1/6 × 1/5 6 × 5 = 30 parts 1/30

Page 11 of 47

Class 7 Maths Chapter 8 Working with Fractions AglaSem · NCERT Solutions

1/15

3 rows × 5 columns = 15 equal parts

The picture for (a): one-third of one-fifth is one cell out of 15, that is 1⁄15.

Why it happens: the horizontal cuts and the vertical cuts are independent, so the
number of small cells is always rows × columns.

Q2 Now, find 1⁄12 × 1⁄18.

Drawing 216 little cells would be painful — use the pattern instead.

Number of rows = 18 (denominator of the multiplicand)

Number of columns = 12 (denominator of the multiplier)

Whole is cut into 18 × 12 = 216 equal parts

1/18 × 1/12 = 1/(18 × 12) = 1/216

In general, for two fractional units:

Page 12 of 47

Class 7 Maths Chapter 8 Working with Fractions AglaSem · NCERT Solutions

1⁄ × 1⁄ = 1/(b × d)
b d

Tip: when a picture becomes clumsy, look at what the picture was doing and turn
that into a rule. That is how every formula in this chapter is born.

Q3 Find the following products. Use a unit square as a whole for representing the
fractions and carrying out the operations. (a) 2⁄3 × 4⁄5 (b) 1⁄4 × 2⁄3 (c) 3⁄5 × 1⁄2 (d) 4⁄6 ×
3⁄
5

Now more than one cell gets shaded — as many as (numerator × numerator).

QUESTION TOTAL PARTS PARTS SHADED PRODUCT

(a) 2/3 × 4/5 3 × 5 = 15 2×4=8 8/15

(b) 1/4 × 2/3 4 × 3 = 12 1×2=2 2/12 = 1/6

(c) 3/5 × 1/2 5 × 2 = 10 3×1=3 3/10

(d) 4/6 × 3/5 6 × 5 = 30 4 × 3 = 12 12/30 = 2/5

Why it happens: the shaded piece is a rectangle 2 cells wide and 4 cells tall inside a
3 × 5 grid — its area is 2 × 4 cells out of 3 × 5. That is Brahmagupta's rule appearing
on its own.

In-text Questions — Pages 181–182
Multiplying Numerators and Denominators; Simplifying to Lowest Form

Q1 Now, find 5⁄12 × 7⁄18.

Follow the same two steps, but with the general numbers.

Page 13 of 47

as e
Class 7 Maths Chapter 8 Working with Fractions
a g l AglaSem · NCERT Solutions

co m
e m.
Whole cut into 18 rows and 12 columns = 12 × 18 parts
m l as
.co
7/18 ÷ 12 = 7/(12 × 18)
m a g
l a se
g
Multiply by 5: (5 × 7)/(12 × 18)
a= 35/216

co m
m . ag
se
5⁄ 7⁄ 35⁄
a
12 × 18 =
l
216

ag (Brāhmasphuṭasiddhānta, 628 CE):
In general — Brahmagupta's formula

co m
m.
a⁄ × c⁄ = (a × c)/(b × d)

as e
b d
m l
.co a g
a s emyou know? The same formula works when one number is a whole number —
gl
Did
a just write it with denominator 1. For example 3 × 3⁄4 = 3⁄1 × 3⁄4 = 9⁄4.

m a s
em
.co agl
a s
Q2 Multiply the following a gl and express the product in its lowest form: 12⁄7 × 5⁄24
fractions

co m
m .
m as e
.co
Cancel the common factor before multiplying.
a g l
a s em
agl 12/7 × 5/24 = (12 × 5)/(7 × 24)
12 and 24 share the factor 12
se m
com g l a
m. a
ase
12 ÷ 12 = 1, 24 ÷ 12 = 2

agl
= (1 × 5)/(7 × 2)

= 5/14

co m
m .
m as e
.co a g l
Why it is allowed: a fraction does not change when its numerator and denominator

se m are divided by the same number. Cancelling first keeps the numbers small —
g l a
a multiplying first would give 60⁄168, which still simplifies to 5⁄14.
c
m .
m a s e
e m . co agl
g l as
a

co m
m .
m ase
.co


a g l Page 14 of 47

Class 7 Maths Chapter 8 Working with Fractions AglaSem · NCERT Solutions

Q3 Let us use the same technique to do one more multiplication. 14⁄15 × 25⁄42

Cancel across the numerators and denominators, then multiply.

14/15 × 25/42 = (14 × 25)/(15 × 42)

14 and 42 share 14 → 1 and 3

25 and 15 share 5 → 5 and 3

= (1 × 5)/(3 × 3)

= 5/9

Did you know? Reducing a fraction to its lowest terms is called apavartana in India.
The Jaina scholar Umasvati (c. 150 CE) used it as a simile even in a philosophical
work.

Figure it Out — Pages 183–184
Section 8.1 Multiplication of Fractions

TRY THIS

Q1 A water tank is filled from a tap. If the tap is open for 1 hour, 7⁄10 of the tank gets
filled. How much of the tank is filled if the tap is open for (a) 1⁄3 hour ____ (b) 2⁄3 hour

____ (c) 3⁄4 hour ____ (d) 7⁄10 hour ____ (e) For the tank to be full, how long should the
tap be running?

Each part is (time) × 7⁄10.

Page 15 of 47

Class 7 Maths Chapter 8 Working with Fractions AglaSem · NCERT Solutions

TAP OPEN FOR WORKING PART OF TANK FILLED

(a) 1/3 hour 1/3 × 7/10 7/30

(b) 2/3 hour 2/3 × 7/10 = 14/30 7/15

(c) 3/4 hour 3/4 × 7/10 21/40

(d) 7/10 hour 7/10 × 7/10 49/100

(e) The tank is full when the filled part is 1.

Time = 1 ÷ 7/10

= 1 × 10/7

= 10/7 hours = 1 3/7 hours ≈ 1 hour 26 minutes

Why it happens: in 7⁄10 of an hour only 49⁄100 of the tank fills — less than half —
because you are taking a part of a part.

Q2 The government has taken 1⁄6 of Somu’s land to build a road. What part of the land
remains with Somu now? She gives half of the remaining part of the land to her
daughter Krishna and 1⁄3 of it to her son Bora. After giving them their shares, she
keeps the remaining land for herself. (a) What part of the original land did Krishna
get? (b) What part of the original land did Bora get? (c) What part of the original
land did Somu keep for herself?

First find what is left after the road is built.

Land left = 1 − 1/6 = 5/6 of the original land

(a) Krishna gets half of that:

1/2 × 5/6 = 5/12

(b) Bora gets one-third of that:

Page 16 of 47

Class 7 Maths Chapter 8 Working with Fractions AglaSem · NCERT Solutions

1/3 × 5/6 = 5/18

(c) Somu keeps the rest:

5/6 − 5/12 − 5/18

LCM of 6, 12, 18 = 36

= 30/36 − 15/36 − 10/36

= 5/36

Check it yourself: 1/6 + 5/12 + 5/18 + 5/36 = 6/36 + 15/36 + 10/36 + 5/36 = 36/36 = 1
✔ All the land is accounted for.

Q3 Find the area of a rectangle of sides 3 3⁄4 ft and 9 3⁄5 ft.

Change both mixed fractions into improper fractions, then multiply.

3 3/4 = 15/4 9 3/5 = 48/5

Area = 15/4 × 48/5

Cancel: 15 and 5 → 3 and 1; 48 and 4 → 12 and 1

= 3 × 12

= 36 square feet

Why it works: the area rule you know for whole-number sides holds for fractional
sides too — that is exactly what the unit-square picture showed.

Page 17 of 47

Class 7 Maths Chapter 8 Working with Fractions AglaSem · NCERT Solutions

Q4 Tsewang plants four saplings in a row in his garden. The distance between two
saplings is 3⁄4 m. Find the distance between the first and last sapling. [Hint: Draw a

rough diagram with four saplings with distance between two saplings as 3⁄4 m]

Four saplings in a row leave 3 gaps, not 4.

3/4 m 3/4 m 3/4 m

1st 4th

Four saplings, three gaps of 3⁄4 m each.

Distance = 3 × 3/4

= 9/4

= 2 1/4 m

Tip: this is the classic “fence-post” trap — for n objects in a row there are only n − 1
gaps.

Q5 Which is heavier: 12⁄15 of 500 grams or 3⁄20 of 4 kg?

Bring both to the same unit — grams.

Page 18 of 47

as e
Class 7 Maths Chapter 8 Working with Fractions
a g l AglaSem · NCERT Solutions

co m
e m.
12/15 of 500 g = (12 × 500)/15
m l as
= 6000/15
m .co a g
l a se
g
= 400 g
a

co m
. ag
4 kg = 4000 g
e m
3/20 of 4000 g = (3 × 4000)/20
g l as
= 12000/20 a

co m
m.
= 600 g

m g) is heavier than 12⁄ of 500 g (400 g), by 200 g. glas e
. c o a
m
3⁄
e
of 4 kg (600

as
20 15

a g l
Why the small fraction wins: 3⁄20 is a much smaller fraction than 12⁄15, but it is

m a s
.co agl
taken of a quantity 8 times bigger. Always check the whole before comparing
fractions.
se m
g l a
a

co m
In-text Questions — Pages 184–186
m .
m as e
.co l
Is the Product Always Greater than the Numbers Multiplied? / Order of Multiplication

a g
a s em
agl Q1 When do you think the product is greater than both the numbers multiplied, when
m
is it in between the two numbers, and when is it smaller than both? [Hint: The

a se
com l
relationship between the product and the numbers multiplied depends on whether
. a g
m
they are between 0 and 1 or they are greater than 1.]

l a se
ANSWER ag
m
Everything depends on where each number sits compared with 1.

. co
em
m l as
.co
SITUATION MULTIPLICATION RELATIONSHIP

a g
se m Both numbers greater than 1
l a
4/3 × 4 = 16/3 Product is greater than both

ag
.c
m
Both numbers between 0 and 1 3/4 × 2/5 = 3/10 Product is less than both

m a s e
. co agl
One between 0 and 1, one greater than 1
m
3/4 × 5 = 15/4 Product lies in between the two

as e
a g l

co m
m .
m ase
.co


a g l Page 19 of 47

Class 7 Maths Chapter 8 Working with Fractions AglaSem · NCERT Solutions

Check the middle row: 3/4 = 15/20 and 2/5 = 8/20

Product 3/10 = 6/20

6/20 < 8/20 < 15/20 ✔

Why it happens: multiplying by a number smaller than 1 means taking only a part of
the other number, so the answer shrinks. Multiplying by a number bigger than 1
means taking more than all of it, so the answer grows.

Q2 Create more such examples for each situation and observe the relationship
between the product and the numbers being multiplied. What can you conclude
about the relationship between the numbers multiplied and the product? Fill in the
blanks: • When one of the numbers being multiplied is between 0 and 1, the product
is ____ (greater/less) than the other number. • When one of the numbers being
multiplied is greater than 1, the product is ____ (greater/less) than the other
number.

Both blanks follow from the same idea.

When one of the numbers is between 0 and 1, the product is less than the other number.
When one of the numbers is greater than 1, the product is greater than the other number.

YOUR OWN EXAMPLE PRODUCT COMPARED WITH THE OTHER NUMBER

1/2 × 9 9/2 = 4 1/2 less than 9

5/2 × 6 15 greater than 6

2/3 × 3/7 2/7 less than both

7/5 × 10/9 14/9 greater than both

Tip: “multiplication always makes things bigger” is only true for whole numbers
greater than 1. With fractions it is simply false.

Page 20 of 47

Class 7 Maths Chapter 8 Working with Fractions AglaSem · NCERT Solutions

Q3 We know that 1⁄2 × 1⁄4 = 1⁄8. Now, what is 1⁄4 × 1⁄2?

It is 1⁄8 too.

1/4 × 1/2 = (1 × 1)/(4 × 2) = 1/8

So a⁄b × c⁄d = c⁄d × a⁄b

Why it happens: a rectangle keeps the same area if you turn it on its side — length
and breadth simply swap places. The same is visible in Brahmagupta's formula,
because a × c = c × a and b × d = d × b.

In-text Questions — Pages 186–188
Section 8.2 Division of Fractions

Q1 What is 12 ÷ 4? You know this already. But can this problem be restated as a
multiplication problem? What should be multiplied by 4 to get 12?

12 ÷ 4 = 3, and the same fact can be written the other way round.

12 ÷ 4 = 3
means 4 × ? = 12

4 × 3 = 12 ✔

Why this matters: every division question is a “missing factor” question. Turning ÷
into × is what lets us divide fractions without inventing a new method.

Q2 What is 1 ÷ 2⁄3?

Ask instead: what times 2⁄3 gives 1?

Page 21 of 47

Class 7 Maths Chapter 8 Working with Fractions AglaSem · NCERT Solutions

2/3 × ? = 1

Turn 2/3 upside down: 2/3 × 3/2 = 6/6 = 1 ✔

So 1 ÷ 2/3 = 3/2

Tip: 3⁄2 is called the reciprocal (व्युत्क्रम) of 2⁄3. A fraction times its reciprocal is always 1.

Q3 Let us try another problem: 3 ÷ 2⁄3. This is the same as 2⁄3 × ? = 3. Can you find the
answer?

We already know what makes 2⁄3 into 1 — now make it 3 times as much.

2/3 × 3/2 = 1

2/3 × (3/2 × 3) = 3

So 3 ÷ 2/3 = 3/2 × 3

= 9/2 = 4 1/2

Why the answer is bigger than 3: you are asking how many two-thirds fit into 3.
Since each piece is smaller than 1, more than 3 of them fit — four and a half, in fact.

Q4 What is 1⁄5 ÷ 1⁄2?

Rewrite as a multiplication and use the reciprocal of the divisor.

1/2 × ? = 1/5

1/2 × 2 = 1, so 1/2 × (2 × 1/5) = 1/5

1/5 ÷ 1/2 = 2 × 1/5

= 2/5

Page 22 of 47

Class 7 Maths Chapter 8 Working with Fractions AglaSem · NCERT Solutions

Check it yourself: 2⁄5 × 1⁄2 = 2⁄10 = 1⁄5 ✔

Q5 What is 2⁄3 ÷ 3⁄5?

The reciprocal of the divisor 3⁄5 is 5⁄3.

3/5 × ? = 2/3

3/5 × 5/3 = 1, so 3/5 × (5/3 × 2/3) = 2/3

2/3 ÷ 3/5 = 5/3 × 2/3
= 10/9 = 1 1/9

Check it yourself: 10⁄9 × 3⁄5 = 30⁄45 = 2⁄3 ✔

Q6 In each of the division problems above, observe how we found the answer. Can we
frame a rule that tells us how to divide two fractions?

Yes — two steps every time.

1. Find the reciprocal of the divisor (turn it upside down).
2. Multiply the dividend by that reciprocal.

a⁄ ÷ c⁄ = a⁄ × d⁄ = (a × d)/(b × c)
b d b c

DIVISION RECIPROCAL OF DIVISOR QUOTIENT

1 ÷ 2/3 3/2 3/2

3 ÷ 2/3 3/2 9/2

1/5 ÷ 1/2 2/1 2/5

2/3 ÷ 3/5 5/3 10/9

Page 23 of 47

as e
Class 7 Maths Chapter 8 Working with Fractions
a g l AglaSem · NCERT Solutions

co m
m.
Did you know? This rule was first stated in general form by Brahmagupta in the

as e
comin the Līlāvatī (1150 CE). l
Brāhmasphuṭasiddhānta (628 CE), and Bhāskara II explained it through the idea of
. a g
em
the reciprocal
a s
agl

co m
ag
In-text Questions — Page 189
m .
Dividend, Divisor and the Quotient
as e
a g l
When do you think the quotient is less than the dividend and when is it greater
m
Q1

co
m.
than the dividend?

m as e
ANSWER co
. a g l
It a s em on the divisor, exactly as with multiplication.
agl
depends

Divisor greater than 1 → quotient is less than the dividend.

m a s
codividend. agl
Divisor between 0 and 1 → quotient is greater than the dividend.
.
em
Divisor equal to 1 → quotient equals the
a s
DIVISION DIVISOR a
gl QUOTIENT COMPARED WITH DIVIDEND

co m
.
6÷3 3>1 2 less than 6

e m
m l as
.co g
6 ÷ 1/4 1/4 < 1 24 greater than 6

e1/8m÷ 1/4 a
a s
gl
1/4 < 1 1/2 greater than 1/8
a
se m
com
Why it happens: dividing by 1⁄4 is the same as multiplying by 4, because the
g l a
m . a
ase
reciprocal of a number below 1 is above 1. So a small divisor makes a big answer.

agl

. com
Is there a similar relationship between the divisor and the quotient? Use your
em the questions
Q2

m a s
understanding of such relationships in multiplication to answer

. co above. a gl
a s em
agl c
.

s e m
m a
Yes — for a fixed dividend, the divisor and the quotient move in opposite directions.

em . co agl
g l as
a

co m
m .
m as e
.co


a g l Page 24 of 47

Class 7 Maths Chapter 8 Working with Fractions AglaSem · NCERT Solutions

12 ÷ 2 = 6

12 ÷ 4 = 3 (divisor doubled → quotient halved)

12 ÷ 1/2 = 24 (divisor made 4 times smaller than 2 → quotient 4 times bigger)

So the bigger the divisor, the smaller the quotient, and the smaller the divisor, the bigger the
quotient. The two are equal only when divisor × divisor = dividend, for example 9 ÷ 3 = 3.

Tip: quotient × divisor = dividend, always. If one of the two factors grows, the other
must shrink to keep the same product.

In-text Questions — Pages 190–191
Section 8.3 Some Problems Involving Fractions

Q1 Example 3: Leena made 5 cups of tea. She used 1⁄4 litre of milk for this. How much
milk is there in each cup of tea?

One-quarter litre of milk is shared equally by 5 cups.

Milk per cup = 1/4 ÷ 5

Reciprocal of 5 is 1/5

= 1/5 × 1/4

= 1/20 litre = 50 ml

Why it happens: 5 × (milk per cup) must be 1⁄4 litre. Twenty such cups would use one
full litre, so each cup holds 1⁄20 litre.

Page 25 of 47

Class 7 Maths Chapter 8 Working with Fractions AglaSem · NCERT Solutions

Q2 Example 4: Here is an example from Baudhāyana’s Śhulbasūtra (c. 800 BCE). Cover
an area of 7 1⁄2 square units with square bricks each of whose sides is 1⁄5 units. How
many such square bricks are needed?

Find the area of one brick first.

Area of one brick = 1/5 × 1/5 = 1/25 square units

Total area = 7 1/2 = 15/2 square units

Number of bricks = 15/2 ÷ 1/25

Reciprocal of 1/25 is 25

= 25 × 15/2

= 375/2 = 187 1/2 bricks

Did you know? The answer really is 187½ — the Śhulbasūtra builders cut bricks in
half while laying fire altars, so half-bricks were perfectly normal for them.

Q3 Example 5: Four fountains fill a cistern. The first fountain can fill the cistern in a
day. The second can fill it in half a day. The third can fill it in a quarter of a day. The
fourth can fill the cistern in one fifth of a day. If they all flow together, in how much
time will they fill the cistern? In a day, the number of times — the first fountain will
fill the cistern is 1 ÷ 1 = 1; the second is 1 ÷ 1⁄2 = ____; the third is 1 ÷ 1⁄4 = ____; the
fourth is 1 ÷ 1⁄5 = ____. The number of times the four fountains together will fill the
cistern in a day is ___ + ___ + ___ + ___ = 12.

Count how many cisterns each fountain fills in one full day.

Page 26 of 47

Class 7 Maths Chapter 8 Working with Fractions AglaSem · NCERT Solutions

FOUNTAIN TIME FOR ONE FILLING FILLINGS IN A DAY

First 1 day 1÷1=1

Second 1/2 day 1 ÷ 1/2 = 2

Third 1/4 day 1 ÷ 1/4 = 4

Fourth 1/5 day 1 ÷ 1/5 = 5

Together in a day = 1 + 2 + 4 + 5 = 12 fillings

Time for 1 filling = 1 ÷ 12

= 1/12 day = 2 hours

Why we add the rates, not the times: the four fountains pour at the same
moment, so their speeds add up. Twelve cisterns a day means one cistern in a twelfth
of a day.

Math Talk — Pages 192–193

Page 27 of 47

Class 7 Maths Chapter 8 Working with Fractions AglaSem · NCERT Solutions

Fractional Relations

MATH TALK

Q1 Here is a square with some lines drawn inside (Fig. 8.4). What fraction of area of the
whole square does the shaded region occupy?

Fig. 8.4, page 192 — the square with lines drawn inside it; the region in question is
hatched.

Peel the picture one layer at a time, taking the whole square as 1 square unit.

Page 28 of 47

as e
Class 7 Maths Chapter 8 Working with Fractions
a g l AglaSem · NCERT Solutions

co m
e m.
Step 1: the top-right square is a quarter of the whole = 1/4 sq unit
m l as
.co
Step 2: the yellow triangle inside it is half of that = 1/2 × 1/4 = 1/8 sq unit
m a g
l a se
g
Step 3: the shaded part is 3/4 of that triangle = 3/4 × 1/8
a= 3/32 square units
o m
. c
em square. ag
s
So the shaded region occupies 3⁄32 of the whole
a
agl
Why this method works: “a fraction of a fraction of a fraction” is just a chain of

co m
m.
multiplications. Each step shrinks the piece you are looking at, and multiplying the
three fractions gives the share of the original whole.
m as e
.co a g l
se m
g l a
a Q2 What fraction of this yellow triangle is shaded? Are you able to see why?

m a s
m .co agl
se

l a
ag
3⁄ of the yellow triangle is shaded.
4

co m
m .
m as e
.co a g l
se m
g l a
a
se m
com g l a
m . a
ase
agl

co m
m .
m as e
.co a g l
se m
g l a
a c
m .
m a s e
em . co agl
g l as
a

co m
m .
m ase
.co


a g l Page 29 of 47

Class 7 Maths Chapter 8 Working with Fractions AglaSem · NCERT Solutions

4

3
1 2

4 equal pieces; pieces 2, 3, 4
are shaded
Joining the midpoints of the two legs to the midpoint of the hypotenuse cuts the triangle into four
equal pieces. Three of them are shaded.

Unshaded piece = 1 of the 4 equal pieces = 1/4

Shaded = 1 − 1/4 = 3/4

Why the four pieces are equal: the vertical line from the midpoint of the base and
the horizontal line from the midpoint of the vertical side both meet the hypotenuse
at its midpoint. They cut the triangle into a small triangle, a square (= two such
triangles) and another small triangle — four congruent halves in all.

Page 30 of 47

Class 7 Maths Chapter 8 Working with Fractions AglaSem · NCERT Solutions

Q3 In each of the figures given below, find the fraction of the big square that the
shaded region occupies.

(a) (b)

The two figures printed with this question on page 193.

Figure (a): 3⁄8. The shaded band lies between two parallel diagonals.

Big triangle below the main diagonal = 1/2 of the square

Small triangle below the shorter parallel line = 1/2 × 1/2 × 1/2 = 1/8

Shaded band = 1/2 − 1/8
= 4/8 − 1/8

= 3/8

Figure (b): 1⁄16. Everything happens inside the top-left quarter.

Top-left quarter = 1/4 of the square

The tilted square joining its midpoints = 1/2 × 1/4 = 1/8

The 4 corner triangles together = 1/4 − 1/8 = 1/8, so each = 1/32

Two corner triangles are shaded = 2 × 1/32

= 1/16

Page 31 of 47

Class 7 Maths Chapter 8 Working with Fractions AglaSem · NCERT Solutions

Tip: in (b) the shaded pieces are the corner triangles on the right of the tilted square
— one at the top, one at the bottom.

In-text Questions — Pages 193–194
A Dramma-tic Donation

Q1 “O wise one! A miser gave to a beggar 1⁄5 of 1⁄16 of 1⁄4 of 1⁄2 of 2⁄3 of 3⁄4 of a dramma. If
you know the mathematics of fractions well, tell me O child, how many cowrie
shells were given by the miser to the beggar.” (1 dramma = 1280 cowrie shells)

“Of” means multiply — so multiply all six fractions.

1/2 × 2/3 × 3/4 × 1/5 × 1/16 × 1/4

1/2 × 2/3 = 1/3

1/3 × 3/4 = 1/4

1/4 × 1/5 = 1/20

1/20 × 1/16 = 1/320

1/320 × 1/4 = 1/1280 of a dramma

1 dramma = 1280 cowrie shells

Gift = 1/1280 × 1280

= 1 cowrie shell

Did you know? Bhāskarāchārya wrapped a joke inside the arithmetic — after all that
grand-sounding fraction talk, the miser parted with a single cowrie, the smallest coin
there was.

Page 32 of 47

Class 7 Maths Chapter 8 Working with Fractions AglaSem · NCERT Solutions

Q2 If we assume 1 gold dinar = 12 silver drammas, 1 silver dramma = 4 copper panas, 1
copper pana = 6 mashakas, and 1 pana = 30 cowrie shells, then 1 copper pana = 1⁄48
gold dinar. 1 cowrie shell = ____ copper panas. 1 cowrie shell = ____ gold dinar.

Work backwards along the chain of coins.

1 pana = 30 cowrie shells

So 1 cowrie shell = 1/30 copper pana

1 pana = 1/48 gold dinar
1 cowrie shell = 1/30 × 1/48

= 1/1440 gold dinar

1 GOLD DINAR = VALUE

in drammas 12 × 1 12 drammas

in panas 12 × 4 48 panas

in cowrie shells 48 × 30 1440 cowries

Why it works: each “1 of the bigger coin = n of the smaller coin” becomes “1 of the
smaller = 1⁄n of the bigger”. Chaining the fractions multiplies them.

Figure it Out — Pages 196–198
Chapter Exercises

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⁄ 1⁄ ; 1⁄ ÷ 11⁄ ; 3 2 ⁄3 ÷ 1 3 ⁄8
5÷ 9 6 12

Turn every divisor upside down and multiply.

Page 33 of 47

ase
Class 7 Maths Chapter 8 Working with Fractions
a g l AglaSem · NCERT Solutions

co m
m.
DIVISION MULTIPLY BY THE RECIPROCAL ANSWER

m as e
3 ÷ 7/9
.co
3 × 9/7
a g l27/7 = 3 6/7

se m
g l a 14/4 × 1/2
a
14/4 ÷ 2 14/8 = 7/4 = 1 3/4

2/3 ÷ 2/3 2/3 × 3/2 1

co m
14/6 ÷ 7/3 14/6 × 3/7
e m . 42/42 = 1 ag
g l as
4/3 ÷ 3/4 4/3 × 4/3
a 16/9 = 1 7/9

m
.co
7/4 ÷ 1/7 7/4 × 7 49/4 = 12 1/4

15em
s
com gla
8/2 ÷ 4/15 4 × 15/4

m . a
ase
1/5 ÷ 1/9 1/5 × 9 9/5 = 1 4/5

agl1/6 ÷ 11/12 1/6 × 12/11 12/66 = 2/11

m a s
.co agl
3 2/3 ÷ 1 3/8 11/3 × 8/11 8/3 = 2 2/3

se m
g l a
a
Tip: change mixed fractions first — 3 2⁄3 = 11⁄3 and 1 3⁄8 = 11⁄8. The 11s then cancel
beautifully.
co m
m .
m as e
.co a g l
a s em For each of the questions below, choose the expression that describes the solution.
gl
Q2

a 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
se m
4 c o m g l a
decorate? (i) 8 × 1⁄ (ii) 1⁄
.
× 1⁄
a
(iii) 8 ÷ 1⁄ (iv) 1⁄ ÷ 8. (b) 1⁄ meter of ribbon is used to

m
se length of the ribbon used for each badge? (i) 8 × ⁄2 (ii) ⁄2
4 8 4 4 2

l a
make 8 badges. What is the 1 1

g
÷ 1⁄8 (iii) 8 ÷ 1⁄2 (iv) 1⁄2a÷ 8. (c) A baker needs 1⁄6 kg of flour to make one loaf of bread.

m
He has 5 kg of flour. How many loaves of bread can he make? (i) 5 × 1⁄6 (ii) 1⁄6 ÷ 5 (iii) 5
co
m .
e
÷ 1⁄6 (iv) 5 × 6
m l as
.co a g
a s emANSWER
agl (a) Option (iii) — how many quarter-metre pieces are in 8 m?
.c
s e m
m a
8 ÷ 1/4 = 8 × 4 = 32 bags
e m . co agl
g l as
a
(b) Option (iv) — half a metre is shared among 8 badges.

co m
m .
m ase
.co


a g l Page 34 of 47

Class 7 Maths Chapter 8 Working with Fractions AglaSem · NCERT Solutions

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?

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⁄ , 1 ÷ 1⁄ , 1 ÷ 1⁄ , 1 ÷ 1⁄ , and 1 ÷ 1⁄ ”. What should the friend say?
6 10 13 9 2

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

Page 35 of 47

Class 7 Maths Chapter 8 Working with Fractions AglaSem · NCERT Solutions

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?

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?

Multiply the mileage by the number of litres.

Page 36 of 47

Class 7 Maths Chapter 8 Working with Fractions AglaSem · NCERT Solutions

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?

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?

Find what is left, then split it three ways.

Page 37 of 47

Class 7 Maths Chapter 8 Working with Fractions AglaSem · NCERT Solutions

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⁄ 707⁄ 707⁄
465 (c) > 676 (d) < 676 (e) > 1 (f) < 1

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?

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

Page 38 of 47

as e
Class 7 Maths Chapter 8 Working with Fractions
a g l AglaSem · NCERT Solutions

co m
e m.
m l as
m .co a g
l a se
a g

co m
em . ag
g l as
a

co m
em.
m l as
m .co a g
l a se
a g
m a s
m .co agl
l a se
ag

co m
m .
m as e
.co a g l
se m
g l a the “Y” splits the
a
bottom-right quarter se m
om c g l a
m . a
a s e
agl
Inside the bottom-right quarter, the shaded part is a triangle plus a square — 3 of its 8 equal halves.

co m
m .
m as e
.co a g l
se m
g l a
a c
m .
m a s e
e m . co agl
g l as
a

co m
m .
m ase
.co


a g l Page 39 of 47

Class 7 Maths Chapter 8 Working with Fractions AglaSem · NCERT Solutions

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.

Page 40 of 47

Class 7 Maths Chapter 8 Working with Fractions AglaSem · NCERT Solutions

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?

Mango tree Sugarcane field

Fig. 8.7, page 198 — the ants’ paths. The group splits equally at every red point.

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

SPLIT GOES TO MANGO TREE CARRIES ON

1st point 1/2 1/2

2nd point 1/4 1/4

3rd point 1/16 + 1/16 (that branch splits again) 1/8 → splits into 1/16 and 1/16

4th point 1/32 1/32 → sugarcane field

Page 41 of 47

Class 7 Maths Chapter 8 Working with Fractions AglaSem · NCERT Solutions

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.

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

Page 42 of 47

Class 7 Maths Chapter 8 Working with Fractions AglaSem · NCERT Solutions

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.

Puzzle Time — Page 199
Chess Puzzles — Non-attacking Queens

TRY THIS

Q1 Chess is a popular 2-player strategy game. This game has its origins in India. It is
played on an 8 × 8 chequered grid. There are 2 sets of pieces — black and white —
one set for each player. Find out how each piece should move and the rules of the
game.

Chess grew out of the ancient Indian game chaturanga. Here is how the six pieces move.

PIECE HOW IT MOVES NUMBER PER
PLAYER

King (राजा) One square in any direction 1

Queen Any number of squares along a row, column or diagonal 1
(वज़ीर)

Rook (हाथी) Any number of squares along a row or column 2

Bishop (ऊँट) Any number of squares along a diagonal 2

Knight (घोड़ा) In an “L”: two squares one way, one square across; it may jump over 2
pieces

Pawn (प्यादा) One square forward (two on its first move); captures diagonally 8

The aim is checkmate — trapping the opponent's King so that it cannot escape capture.

Did you know? The Queen is the strongest piece precisely because it combines the
Rook's and the Bishop's moves — which is what makes the puzzle below hard.

Page 43 of 47

as e
Class 7 Maths Chapter 8 Working with Fractions
a g l AglaSem · NCERT Solutions

co m
m.
From its current position, a Queen piece can move along the horizontal, vertical or
e
Q2

m l as
.co
diagonal. Place 4 Queens such that no 2 queens attack each other.

a g
se m
g l a
a

On a 4 × 4 board there must be exactly one Queen in every row and every column, and no two

m
on a diagonal.
. co ag
e m
g l as
a

co m
m.

ase
m .co
m
Q a g l as e

agl
s

g l a se m .co
m
Q agl
a

a

co m

m .co
m Q a g l as e m .

l a se
a g

s e m . com Q a g l a se m

a
agl
rows 1–4 → columns . om 4,
c2,
m a s em
e m . co 1, 3 agl
g l as
a c
m .
m a s e
co agl
One valid arrangement of 4 non-attacking Queens on a 4 × 4 board.

m .
as e
a g l

co m
m .
m ase
.co


a g l Page 44 of 47

Class 7 Maths Chapter 8 Working with Fractions AglaSem · NCERT Solutions

Row 1 → column 2

Row 2 → column 4

Row 3 → column 1

Row 4 → column 3

Why it works: all four columns are different, so no vertical attack. For diagonals,
check row + column: 3, 6, 4, 7 — all different; and row − column: −1, −2, 2, 1 — all
different. Equal sums or equal differences would mean a shared diagonal.

Q3 Now, place 8 queens on this 8 × 8 grid so that no 2 queens attack each other!

Again, one Queen per row and per column, with all diagonals clear.

ROW 1 2 3 4 5 6 7 8

COLUMN 1 5 8 6 3 7 2 4

row + column: 2, 7, 11, 10, 8, 13, 9, 12 — all different ✔

row − column: 0, −3, −5, −2, 2, −1, 5, 4 — all different ✔

So no two Queens share a row, a column or a diagonal.

Did you know? The 8-queens puzzle has 92 solutions altogether, but only 12 of them
are really different — the rest are rotations and mirror images of those 12.

Chapter at a glance
Multiplying by a fraction is done in two steps: divide the multiplicand by the multiplier's
denominator, then multiply by its numerator.
Drawing the two fractions as the sides of a rectangle inside a unit square shows why area =
product of the sides.

Page 45 of 47

Class 7 Maths Chapter 8 Working with Fractions AglaSem · NCERT Solutions

Brahmagupta's formula: a⁄b × c⁄d = a × c⁄b × d. Common factors may be cancelled before
multiplying.
The product is not always bigger: multiplying by a number between 0 and 1 makes it
smaller; multiplying by a number greater than 1 makes it bigger.
The reciprocal of a⁄b is b⁄a; a fraction times its reciprocal is 1. To divide, multiply by the
divisor's reciprocal.
Dividing by a number between 0 and 1 makes the quotient larger than the dividend —
which is why 6 ÷ 1⁄4 = 24.

Page 46 of 47

Class 7 Maths Chapter 8 Working with Fractions AglaSem · NCERT Solutions

Quick revision

IDEA WHAT IT MEANS EXAMPLE FROM THE KEY RESULT
CHAPTER

Whole number × Repeated addition of the fraction 3 × 1/4 3/4
fraction

Fraction × whole Divide by the denominator, multiply by 2/5 × 3 6/5
number the numerator

Unit square model The whole is cut into rows × columns 1/2 × 1/4 1/8

Fractional units 1 over the product of denominators 1/b × 1/d 1/(b × d)

Brahmagupta's Numerators × numerators, 5/12 × 7/18 35/216
product rule denominators × denominators

Cancelling Divide out common factors first 12/7 × 5/24 5/14
(apavartana)

Area of a rectangle Fractional sides, area is their product sides 3¾ ft and 9⅗ ft 36 sq ft

Size of the product Depends on whether the numbers are 3/4 × 2/5 3/10, less than
below or above 1 both

Reciprocal (व्युत्क्रम) Turn the fraction upside down 3/5 5/3, and 3/5 ×
5/3 = 1

Division rule Multiply the dividend by the divisor's 2/3 ÷ 3/5 10/9
reciprocal

Small divisor Dividing by a fraction below 1 makes 6 ÷ 1/4 24
the answer bigger

Order does not matter The rectangle is the same if you swap 1/2 × 1/4 = 1/4 × 1/2 1/8
its sides

Page 47 of 47