NCERT Solutions Class 6 Maths Chapter 7 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 6 · M AT H S

NCERT Solutions

Chapter 7: Fractions

NCERT Textbook — Ganita Prakash

BOOK PAGES SECTIONS QUESTIONS MEDIUM

151 – 186 27 77 English

Solutions, notes, sample papers & more at 50 pages

Class 6 Maths Chapter 7 Fractions AglaSem · NCERT Solutions

CLASS 6 · MATHS · GANITA PRAKASH

NCERT Solutions — Chapter 7: Fractions
Complete, step-by-step NCERT Solutions for Class 6 Maths Chapter 7 Fractions from the NCERT textbook
Ganita Prakash. Every Figure it Out question from pages 152 to 182 is solved, along with all the in-text work
— equal shares of roti and chikki, fractional units, fractions on the number line, mixed numbers, the fraction
wall, equivalent fractions, lowest terms, comparing fractions, and Brahmagupta’s method for adding and
subtracting fractions.

TEXTBOOK BOOK PAGES

Ganita Prakash (Class 6) 151 – 186

SECTIONS QUESTIONS

27 77

MEDIUM

English

In-text Questions — Pages 151 – 153
Section 7.1 Fractional Units and Equal Shares

Q1 If one roti is equally shared among 4 children, how much roti will one child get? And
which is more — 1⁄2 roti or 1⁄4 roti?

One roti shared equally by 4 children gives each child 1⁄4 roti.

1 roti ÷ 2 children = 1/2 roti each

1 roti ÷ 4 children = 1/4 roti each

So 1/2 > 1/4

Answer: 1⁄2 roti is more than 1⁄4 roti.

Why it happens: the roti is the same in both cases. In the second group more
children share that same one roti, so each child’s piece must be smaller.

Page 1 of 50

Class 6 Maths Chapter 7 Fractions AglaSem · NCERT Solutions

Q2 Which fraction is greater — 1⁄5 or 1⁄9?

1⁄ 1⁄
5 is greater. Arvin’s guess (“9 is bigger than 5, so 9 is bigger”) is a common mistake.

Think of the fractions as shares:

1 roti shared by 5 children → each gets 1⁄5 roti.

1 roti shared by 9 children → each gets 1⁄9 roti.

More people sharing → smaller share

So 1/9 < 1/5

Tip: for fractional units, the bigger the denominator, the smaller the fraction. That is
why 1⁄100 is bigger than 1⁄200.

Q3 Find out and discuss the words for fractions that are used in the different
languages spoken in your home, city, or state. Ask your grandparents, parents,
teachers, and classmates what words they use for different fractions, such as for
one and a half, three quarters, one and a quarter, half, quarter, and two and a half,
and write them here:

This is a talking-and-collecting activity. Here are the words most families in India use — add the
ones spoken in your own home.

FRACTION IN NUMBERS COMMON INDIAN WORDS

quarter 1/4 पाव (Hindi), कालु (kaal, Tamil), पाव (Marathi)

half 1/2 आधा (Hindi), अध (Marathi), অেধক (Bengali), arai (Tamil)

three quarters 3/4 तीन पाव / पौना (Hindi), मुक्काल (mukkaal, Tamil), tri-pada (Rig Veda)

one and a quarter 1 1/4 सवा (Hindi/Marathi)

one and a half 1 1/2 डेढ़ (Hindi), दीड (Marathi), দড় (Bengali)

two and a half 2 1/2 ढाई (Hindi), अडीच (Marathi), আড় াই (Bengali)

Page 2 of 50

Class 6 Maths Chapter 7 Fractions AglaSem · NCERT Solutions

Did you know? In the Rig Veda the fraction 3⁄4 is called tri-pada — the very same
idea as “teen paav” in everyday Hindi and “mukkaal” in Tamil. Our words for fractions
are thousands of years old.

Figure it Out — Pages 152 & 153
Section 7.1 Fractional Units and Equal Shares

MATH TALK

Q1 Three guavas together weigh 1 kg. If they are roughly of the same size, each guava
will roughly weigh ____ kg.

One kilogram is shared equally by 3 guavas.

1 kg ÷ 3 = 1/3 kg

Answer: each guava weighs about 1⁄3 kg.

Check it yourself: 1/3 + 1/3 + 1/3 = 3/3 = 1 kg ✔

Q2 A wholesale merchant packed 1 kg of rice in four packets of equal weight. The
weight of each packet is ___ kg.

1 kg ÷ 4 = 1/4 kg

Answer: each packet weighs 1⁄4 kg (250 g).

Why 1⁄4: one whole kilogram is cut into 4 equal parts, so each part is one fractional
unit of size 1⁄4.

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as e
Class 6 Maths Chapter 7 Fractions
a g l AglaSem · NCERT Solutions

co m
m.
Four friends ordered 3 glasses of sugarcane juice and shared it equally among
e
Q3

m l as
.co
themselves. Each one drank ____ glass of sugarcane juice.

a g
se m
g l a
a

3 glasses are shared equally by 4 friends.

co m
e m . ag
as
3 ÷ 4 = 3/4 glass

a g l
Answer: each friend drank 3⁄4 glass.

. com
m a s em and 12 ÷
Why it works: cut every glass into 4 quarters. There are 3 × 4 = 12 quarters,
. co for each friend, that is 3⁄4 glass. a gl
em
4 = 3 quarters
a s
a gl
m a s
.co agl
Q4 The big fish weighs 1⁄2 kg. The small one weighs 1⁄4 kg. Together they weigh ____ kg.

se m
g l a
ANSWER a
Measure both fish with the same fractional unit, 1⁄4 kg.
co m
m .
m as e
.co
1/2 = 2/4
a g l
s m + 1/4 = 3 times 1/4 = 3/4 kg
e2/4
gl a
a
se m
Answer: together they weigh 3⁄4 kg (750 g).
com g l a
m . a
gl ase
a
Arrange these fraction words in order of size from the smallest to the biggest in the
Q5

co m
empty box below: One and a half, three quarters, one and a quarter, half, quarter,

m .
e
two and a half.
m l as
.co
mANSWER a g
s e
agla First write each word as a number, then order them.
.c
s e m
m a
e m . co agl
g l as
a

co m
m .
m ase
.co


a g l Page 4 of 50

Class 6 Maths Chapter 7 Fractions AglaSem · NCERT Solutions

WORD FRACTION POSITION

quarter 1/4 smallest

half 1/2 2nd

three quarters 3/4 3rd

one and a quarter 1 1/4 4th

one and a half 1 1/2 5th

two and a half 2 1/2 biggest

1/4 < 1/2 < 3/4 < 1 1/4 < 1 1/2 < 2 1/2

Answer: quarter, half, three quarters, one and a quarter, one and a half, two and a half.

Tip: the first three are less than 1 whole, the last three are more than 1 whole — that
split makes the ordering easy.

In-text Questions — Pages 154 & 155

Page 5 of 50

Class 6 Maths Chapter 7 Fractions AglaSem · NCERT Solutions

Section 7.2 Fractional Units as Parts of a Whole

MATH TALK

Q1 A picture of the chikki broken into 2 pieces is shown below. How much of the
original chikki is each piece?

1
4
A whole chikki

The whole chikki, and the same chikki broken into 2 pieces.

The whole chikki is thought of as 4 equal quarters. The break gives one small piece and one big
piece.

Small piece = 1 quarter = 1/4 chikki

Big piece = 3 quarters = 3 times 1/4 = 3/4 chikki

Check it yourself: 1/4 + 3/4 = 4/4 = 1 whole chikki ✔

Q2 By dividing the whole chikki into 6 equal parts in different ways, we get 1⁄6 chikki
pieces of different shapes. Are they of the same size?

Yes — they are the same size, even though their shapes are different.

Page 6 of 50

Class 6 Maths Chapter 7 Fractions AglaSem · NCERT Solutions

In the first picture the chikki is cut into 6 rectangles; in the second it is cut into 6 triangles. In
both cases the whole chikki has been shared into 6 equal parts, so one part is 1⁄6 chikki either
way.

The big idea: a fraction measures how much, not what shape. Six equal pieces of the
same chikki must each be 1⁄6 of it, whether they are square, rectangular or triangular.

Try This: fold a square sheet into 6 equal parts in two different ways. Cut them and
place one piece of each kind on a weighing balance — they balance.

Q3 What is the fractional unit of chikki shown below?

A whole chikki

A whole chikki, and the piece shown in the book.

The whole chikki has been broken into 3 equal pieces and one of them is shown.

1 whole ÷ 3 equal pieces → each piece = 1/3 chikki

Answer: the fractional unit is 1⁄3.

Figure it Out — Page 155

Page 7 of 50

Class 6 Maths Chapter 7 Fractions AglaSem · NCERT Solutions

Section 7.2 Fractional Units as Parts of a Whole

Q1 The figures below show different fractional units of a whole chikki. How much of a
whole chikki is each piece?

A whole chikki
a. b. c. d. e.

f. g. h.

The eight pieces a–h, with a whole chikki given for comparison.

Look at the whole chikki first. It is a 6 × 4 grid, so it is made of 24 equal small squares. Now just

Page 8 of 50

as e
Class 6 Maths Chapter 7 Fractions
a g l AglaSem · NCERT Solutions

count how many small squares each piece covers.
co m
e m.
m l as
m .co a g
l a se
a g

co m
e m . ag
g l as
a

co m
em.
m l as
m .co a g
l a se
a g
m a s
em
.co agl
a s
agl = 6 × 4 = 24 small squares
A whole chikki
The whole chikki drawn as 24 equal small squares. One small square is 1⁄24 of the chikki.

com
m .
m as e
PIECE
.co HOW MUCH IT COVERS
a g l
FRACTION OF THE CHIKKI

a s ea.m1 × 2 rectangle
agl 2 small squares 2/24 = 1/12

se m
com a
b. big triangle, legs 3 and 4 ½ × 3 × 4 = 6 squares 6/24 = 1/4

g l
½ × .3 × 2 = 3 squares a
e m
as
c. flat triangle, legs 3 and 2 3/24 = 1/8

d. 1 × 4 strip agl 4 small squares 4/24 = 1/6

co m
.
e. L-shape 3 small squares 3/24 = 1/8

e m
m l as
.co g
f. triangle, base 2 and height 4 ½ × 2 × 4 = 4 squares 4/24 = 1/6

em g. single small square a
a s
gl
1 small square 1/24

a c
m .
e
h. small triangle, legs 1 and 2 ½ × 1 × 2 = 1 square 1/24

m a s
e m . co
Answers: a. 1⁄12 b. 1⁄4 c. 1⁄8 d. 1⁄6 e. 1⁄8 f. 1⁄6 g. 1⁄24 h. 1⁄24 agl
g l as
a

co m
m .
m ase
.co


a g l Page 9 of 50

Class 6 Maths Chapter 7 Fractions AglaSem · NCERT Solutions

Why counting squares works: every piece is compared with the same whole. If a
piece covers k of the 24 small squares, it is k⁄24 of the chikki — then write that in
lowest terms.

In-text Questions — Pages 156 & 157
Section 7.3 Measuring Using Fractional Units

Q1 Fold the strip into two equal parts and then open up the strip again. Taking the
strip to be one unit in length, what are the lengths of the two new parts of the strip
created by the crease?

The crease cuts the one-unit strip into 2 equal parts.

Each part = 1 ÷ 2 = 1/2 unit

Check: 1/2 + 1/2 = 2/2 = 1 unit ✔

Answer: both parts are 1⁄2 unit long.

Q2 What will you get if you fold the previously-folded strip again into two equal parts?

Each half is folded into two, so the strip now has 4 equal parts and each part is 1⁄4 unit.

2 times 1/4 = 2/4

3 times 1/4 = 3/4

4 times 1/4 = 4/4 = 1

Why: folding into two doubles the number of parts each time — 1 → 2 → 4 → 8 — so
the fractional unit becomes half as long at every fold.

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Class 6 Maths Chapter 7 Fractions AglaSem · NCERT Solutions

Q3 Do it once more! Fill in the blank boxes. (2 times 1⁄8 = __, 4 times 1⁄8 = __, 6 times 1⁄8 =
__, 8 times 1⁄8 = __)

One more fold gives 8 equal parts, so the fractional unit is now 1⁄8.

2 times 1/8 = 2/8 = 1/4

4 times 1/8 = 4/8 = 1/2

6 times 1/8 = 6/8 = 3/4

8 times 1/8 = 8/8 = 1

Did you notice? The folded strip shows equivalent fractions all by itself: 2⁄8 = 1⁄4, 4⁄8 =
1⁄ , 6⁄ = 3⁄ .
2 8 4

Figure it Out — Page 158
Section 7.3 Measuring Using Fractional Units

Q1 Continue this table of 1⁄2 for 2 more steps.

1⁄ 1⁄ + 1⁄ 1⁄ + 1⁄ + 1⁄ 1⁄ + 1⁄ + 1⁄ + 1⁄ + 1⁄ + 1⁄ +
2 2 2 2 2 2 2 2 2 2 2 2
= 1 times half = 2 times half = 3 times half 1⁄ 1⁄ + 1⁄
2 2 2
= 4 times half = 5 times half

The table of one-half printed in the book.

The table in the book ends at “5 times half”. The next two steps add one more half each time.

Page 11 of 50

Class 6 Maths Chapter 7 Fractions AglaSem · NCERT Solutions

1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 = 6 times 1/2 = 6/2 = 3

1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 = 7 times 1/2 = 7/2 = 3 1/2

Tip: two halves make one whole, so “6 times half” is exactly 3 whole rotis and “7
times half” is 3 rotis and one half.

Q2 Can you create a similar table for 1⁄4?

Yes. Keep adding one quarter at a time.

NUMBER OF QUARTERS ADDITION STATEMENT FRACTION

1 time quarter 1/4 1/4

2 times quarter 1/4 + 1/4 2/4 = 1/2

3 times quarter 1/4 + 1/4 + 1/4 3/4

4 times quarter 1/4 + 1/4 + 1/4 + 1/4 4/4 = 1

5 times quarter 1/4 + 1/4 + 1/4 + 1/4 + 1/4 5/4 = 1 1/4

6 times quarter six quarters 6/4 = 1 1/2

Q3 Make 1⁄3 using a paper strip. Can you use this to also make 1⁄6?

Yes. First fold the strip into three equal parts (roll it like a letter) — each part is 1⁄3 of the strip.

Now fold that folded strip once more into two. Each third gets cut into 2 equal pieces, so the
whole strip now has 3 × 2 = 6 equal pieces.

Half of 1/3 = 1/6

So 2 pieces of 1/6 = 1/3, and 6 pieces of 1/6 = 1

Page 12 of 50

Class 6 Maths Chapter 7 Fractions AglaSem · NCERT Solutions

Why it works: halving a fractional unit doubles the number of parts in the whole —
thirds become sixths, quarters become eighths, and so on.

Q4 Draw a picture and write an addition statement as above to show: a. 5 times 1⁄4 of a
roti b. 9 times 1⁄4 of a roti

(a) 5 times 1⁄4 of a roti — draw 1 full roti cut into 4 quarters, and one more roti with 1 quarter
shaded.

1/4 + 1/4 + 1/4 + 1/4 + 1/4 = 5/4 = 1 + 1/4 = 1 1/4 rotis

(b) 9 times 1⁄4 of a roti — draw 2 full rotis cut into quarters, and a third roti with 1 quarter
shaded.

1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 = 9/4 = 2 + 1/4 = 2 1/4 rotis

Tip: group the quarters in fours — every group of 4 quarters makes 1 whole roti. 9
quarters = 4 + 4 + 1, so 2 whole rotis and one quarter.

Q5 Match each fractional unit with the correct picture: 1⁄3, 1⁄5, 1⁄8, 1⁄6

The four pictures given in the book.

Count the number of equal parts in each circle — one shaded part is 1 divided by that number.

Page 13 of 50

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Class 6 Maths Chapter 7 Fractions
a g l AglaSem · NCERT Solutions

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

com
m . ag
a 1/6 se
8 parts → 1/8 gl→
6 parts
a 3 parts → 1/3 5 parts → 1/5

co m
m.
The four circles of the book, in the order they are printed. The shaded part of each circle is its

m as e
.co l
fractional unit.

a g
a s em
a glFRACTIONAL UNIT PICTURE EQUAL PARTS IN THE CIRCLE

a s
com agl
1/3 third circle 3

m .
ase
1/5
l
fourth circle 5

1/8 agfirst circle 8

co m
.
1/6 second circle 6

se m
o m l a
g cut lines through the
m .cdo not count only the cut lines — count the pieces. Three
a
asecentre make 6 pieces, not 3.
Tip:

agl
se m
com g l a
. a
sem
In-text Questions — Page 159
a
agl
Section 7.4 Marking Fraction Lengths on the Number Line

co m
m
Here, the fractional unit is dividing a length of 1 unit into three equal parts. Write
.
e
Q1

m l as
.co
the fraction that gives the length of the blue line in the box or in your notebook.

a g
a s emANSWER
agl
.c
m
The distance from 0 to 1 is cut into 3 equal parts, so one step is 1⁄3 unit. The blue line covers two

m a s e
co agl
such steps — it ends at the second mark after 0.

m .
ase
2 times 1/3 = 2/3 unit
a g l

com
m .
m ase
.co


a g l Page 14 of 50

Class 6 Maths Chapter 7 Fractions AglaSem · NCERT Solutions

Answer: the blue line is 2⁄3 unit long.

Q2 Here, a unit is divided into 5 equal parts. Write the fraction that gives the length of
the blue lines in the respective boxes or in your notebook.

Now each step is 1⁄5 unit. Count the steps covered by each blue line.

Shorter blue line = 2 steps = 2/5 unit

Longer blue line = 4 steps = 4/5 unit

So the two boxes (after 1⁄5 and after 3⁄5) hold 2⁄5 and 4⁄5.

Q3 Now, a unit is divided into 8 equal parts. Write the appropriate fractions in your
notebook.

Each step is now 1⁄8 unit, so the marks between 0 and 1 are:

1/8, 2/8, 3/8, 4/8, 5/8, 6/8, 7/8, 8/8 = 1

A line that reaches the third mark is 3⁄8 unit long, one that reaches the sixth mark is 6⁄8 unit long,
and so on. Going past 1 the marks continue 9⁄8, 10⁄8, 11⁄8, …

Did you notice? Some of these have shorter names: 2⁄8 = 1⁄4, 4⁄8 = 1⁄2, 6⁄8 = 3⁄4.

Figure it Out — Page 160

Page 15 of 50

Class 6 Maths Chapter 7 Fractions AglaSem · NCERT Solutions

Section 7.4 Marking Fraction Lengths on the Number Line

MATH TALK

Q1 On a number line, draw lines of lengths 1⁄10, 3⁄10, and 4⁄5.

Divide the length from 0 to 1 into 10 equal parts. Then 1⁄10 is 1 step, 3⁄10 is 3 steps, and 4⁄5 = 8⁄10
is 8 steps.

4/5 = (4 × 2)/(5 × 2) = 8/10

1/10
3/10
4/5 = 8/10

0 1

The unit from 0 to 1 cut into ten equal steps. The three coloured lines are 1⁄10, 3⁄10 and 4⁄5 (= 8⁄10) long.

Q2 Write five more fractions of your choice and mark them on the number line.

Any five fractions may be chosen. Here is one good set, marked with the same tenths scale.

FRACTION WHERE IT GOES IN TENTHS

1/2 exactly halfway between 0 and 1 5/10

2/5 2 steps of one-fifth 4/10

7/10 7 steps of one-tenth 7/10

6/5 just past 1 12/10

3/2 halfway between 1 and 2 15/10

Page 16 of 50

Class 6 Maths Chapter 7 Fractions AglaSem · NCERT Solutions

Math Talk: compare your five fractions with a friend’s. Write each one as tenths (or
twentieths) and you can instantly say whose fraction is bigger.

Q3 How many fractions lie between 0 and 1? Think, discuss with your classmates, and
write your answer.

There are unlimited (infinitely many) fractions between 0 and 1.
Take any denominator you like and you already get new fractions:

halves → 1/2

thirds → 1/3, 2/3

tenths → 1/10, 2/10, …, 9/10

hundredths → 1/100, 2/100, …, 99/100

Why they never run out: pick any two fractions between 0 and 1. Their “midpoint” is
again a fraction lying between them — for example between 1⁄2 and 1⁄3 lies 5⁄12. You
can keep doing this forever, so no one can ever finish listing them.

Page 17 of 50

Class 6 Maths Chapter 7 Fractions AglaSem · NCERT Solutions

Q4 What is the length of the blue line and black line shown below? The distance
between 0 and 1 is 1 unit long, and it is divided into two equal parts. The length of
each part is 1⁄2. So the blue line is 1⁄2 units long. Write the fraction that gives the
length of the black line in the box.

0 1 1 2
2

The number line with the blue line and the black line drawn above it.

Each step on this line is 1⁄2 unit. The black line starts at 0 and stops at the mark between 1 and 2,
that is at the third half-step.

Blue line = 1 step = 1/2 unit

Black line = 3 steps = 3 times 1/2 = 3/2 unit = 1 1/2 units

Answer: the black line is 3⁄2 (one and a half) units long.

Q5 Write the fraction that gives the lengths of the black lines in the respective boxes.

Here the unit is divided into 5 equal parts, so every step is 1⁄5 unit. All four black lines start at 0
and end after 1, at the four empty boxes.

Page 18 of 50

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Class 6 Maths Chapter 7 Fractions
a g l AglaSem · NCERT Solutions

co m
m.
se 9/5
m l a
m .co ag
l a se 8/5

a g 6/5
7/5

co m
e m . ag
0
g l as 1 2
a
m
The four black lines of the book redrawn: they stop 6, 7, 8 and 9 fifth-steps away from 0.

co
e m.
co=m6 steps = 6/5 g l as
. a
em box = 7 steps = 7/5
First box
a s
a gl Second

s
Third box = 8 steps = 8/5
m a
Fourth box = 9 steps = 9/5
m .co agl
l a se
a g
Answer: 6⁄5, 7⁄5, 8⁄5, 9⁄5 (that is 11⁄5, 12⁄5, 13⁄5, 14⁄5).

co m
m .
as e
om
In-text.cQuestions — Page 161
a g l
se m
g l a
Section 7.5 Mixed Fractions
a
se m
m l a
Write down all the fractions you marked on the number line earlier. Now, let us
o
Q1

m .c less than 1 unit, and lengths more than 1 unit. ag
classify these in two groups: lengths

l a se
ag
m
LENGTHS LESS THAN 1 UNIT LENGTHS MORE THAN 1 UNIT

. co
e m
m l as
m .co a g
l a se
ag
.c
m
The two groups given in the book.

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

Take all the fractions from pages 159 and 160 and sort them.

co m
m .
m ase
.co


a g l Page 19 of 50

Class 6 Maths Chapter 7 Fractions AglaSem · NCERT Solutions

LENGTHS LESS THAN 1 UNIT LENGTHS MORE THAN 1 UNIT

1/2, 2/3, 2/5, 4/5, 1/10, 3/10, 3/8 3/2, 6/5, 7/5, 8/5, 9/5

The blue lines of the book are all in the first column, the black lines all in the second.

Q2 Did you notice something common between the fractions that are greater than 1?

Yes — look at the two numbers of each fraction.

Less than 1 unit → the numerator is smaller than the denominator (2⁄5, 3⁄8).

More than 1 unit → the numerator is larger than the denominator (7⁄5, 3⁄2).

Exactly 1 → numerator and denominator are equal (5⁄5, 8⁄8).

Why: it takes 5 fifths to make one whole. So 7⁄5 means 5 fifths (one whole) and 2
fifths more — which must be more than 1.

Figure it Out — Page 162
Section 7.5 Fractions greater than one

MATH TALK

Q1 How many whole units are there in 7⁄2?

Two halves make one whole, so group the halves in pairs.

7/2 = (1/2 + 1/2) + (1/2 + 1/2) + (1/2 + 1/2) + 1/2

= 1 + 1 + 1 + 1/2 = 3 + 1/2

Answer: there are 3 whole units in 7⁄2 (and one extra half).

Page 20 of 50

Class 6 Maths Chapter 7 Fractions AglaSem · NCERT Solutions

Q2 How many whole units are there in 4⁄3 and in 7⁄3?

Three thirds make one whole, so group the thirds in threes.

4/3 = (1/3 + 1/3 + 1/3) + 1/3 = 1 whole + 1/3

7/3 = (3 thirds) + (3 thirds) + 1/3 = 2 wholes + 1/3

Answer: 4⁄3 has 1 whole unit; 7⁄3 has 2 whole units.

Tip: the number of whole units is simply “how many times the denominator fits into
the numerator”. 3 fits once in 4, and twice in 7.

Figure it Out — Page 162
Writing fractions greater than one as mixed numbers

Q1 Figure out the number of whole units in each of the following fractions: a. 8⁄3 b. 11⁄5
c. 9⁄4

a. 8/3 → 3 + 3 + 2 thirds → 2 whole units and 2/3 left, 8/3 = 2 2/3

b. 11/5 → 5 + 5 + 1 fifths → 2 whole units and 1/5 left, 11/5 = 2 1/5

c. 9/4 → 4 + 4 + 1 quarters → 2 whole units and 1/4 left, 9/4 = 2 1/4

Answer: each of them contains 2 whole units.

Q2 Can all fractions greater than 1 be written as such mixed numbers?

Almost all — but not the ones that are exactly whole numbers.
A mixed number needs two parts: a whole part and a fractional part that is less than 1. If the
denominator divides the numerator exactly, nothing is left over.

Page 21 of 50

Class 6 Maths Chapter 7 Fractions AglaSem · NCERT Solutions

8/4 = 2 → only a whole part, so it is written simply as 2

9/4 = 2 + 1/4 → written as 2 1/4 ✔

So: every fraction greater than 1 whose numerator is not a multiple of its denominator can be
written as a mixed number; the rest are whole numbers already.

Q3 Write the following fractions as mixed fractions (e.g., 9⁄2 = 41⁄2): a. 9⁄2 b. 9⁄5 c. 21⁄19 d.
47⁄ 12⁄ 19⁄
9 e. 11 f. 6

Divide the numerator by the denominator. The quotient is the whole part and the remainder
becomes the new numerator.

FRACTION DIVISION MIXED NUMBER

a. 9/2 9 ÷ 2 = 4 remainder 1 4 1/2

b. 9/5 9 ÷ 5 = 1 remainder 4 1 4/5

c. 21/19 21 ÷ 19 = 1 remainder 2 1 2/19

d. 47/9 47 ÷ 9 = 5 remainder 2 5 2/9

e. 12/11 12 ÷ 11 = 1 remainder 1 1 1/11

f. 19/6 19 ÷ 6 = 3 remainder 1 3 1/6

Check it yourself: 52⁄9 = 5 × 9 + 2 = 47 ninths = 47⁄9 ✔

Figure it Out — Page 163

Page 22 of 50

Class 6 Maths Chapter 7 Fractions AglaSem · NCERT Solutions

Writing a mixed number as a regular fraction

MATH TALK

Q1 Write the following mixed numbers as fractions: a. 31⁄4 b. 72⁄3 c. 94⁄9 d. 31⁄6 e. 23⁄11 f.
39⁄10

Use Jaya’s idea: change every whole into fractional units, then count all the units.

3 1/4 = (4 × 1/4) + (4 × 1/4) + (4 × 1/4) + (3 × 1/4)

= 4 + 4 + 4 + 3 quarters = 15/4

The short rule is (whole × denominator) + numerator, kept over the same denominator.

MIXED NUMBER WORKING FRACTION

a. 3 1/4 (3 × 4) + 1 = 13 13/4

b. 7 2/3 (7 × 3) + 2 = 23 23/3

c. 9 4/9 (9 × 9) + 4 = 85 85/9

d. 3 1/6 (3 × 6) + 1 = 19 19/6

e. 2 3/11 (2 × 11) + 3 = 25 25/11

f. 3 9/10 (3 × 10) + 9 = 39 39/10

Why multiply by the denominator: one whole roti contains exactly denominator-
many fractional units — 4 quarters, 3 thirds, 10 tenths. So 3 wholes contain 3 × 10 =
30 tenths, and 9 more tenths make 39 tenths.

In-text Questions — Page 164

Page 23 of 50

as e
Class 6 Maths Chapter 7 Fractions
a g l AglaSem · NCERT Solutions

Section 7.6 Equivalent Fractions · the fraction wall
co m
e m.
m l as
Q1
.co
Are the lengths 1⁄2 and 2⁄4 equal? Are the lengths 2⁄4 and 4⁄8 equal?
m a g
l a se
a g

m
Yes to both. Place the three shaded strips one below the other — all three shaded portions end

. co ag
m
at exactly the same place.

as e
a g l
1/2 = 2/4 = 4/8

co m
em.
m effects cancel, so the total length does not change. l as
Why: when you cut every part into two, you get twice as many parts but each is half
c o g
m .
as long. Two a
as e
agl
m a s
.co agl
Q2 Now, check whether 1⁄3 and 2⁄6 are equivalent fractions or not, using paper strips.

se m
g l a
a
Take two identical strips. Fold one into 3 equal parts and shade one part; fold the other into 6

co m
.
equal parts and shade two parts. Put them side by side — the shaded lengths match.

e m
m
=o(1 × 2)/(3 × 2) = 2/6
1/3.c g l as
m a
ase
agl Yes, 1⁄ and 2⁄ are equivalent fractions.
3 6
se m
com g l a
m . a
ase
Q3 agl
Are the lengths 1⁄2 and 3⁄6 equal?

co m
m .
as e
com l
Yes. On the fraction wall the strip of one half and the three strips of one sixth end at the same
. a g
sem
line.
a
agl c
m .
m a s e
e m . co agl
g l as
a

co m
m .
m ase
.co


a g l Page 24 of 50

Class 6 Maths Chapter 7 Fractions AglaSem · NCERT Solutions

1

1/2 1/2

1/3 1/3 1/3

1/4 1/4 1/4 1/4

1/5 1/5 1/5 1/5 1/5

1/6 1/6 1/6 1/6 1/6 1/6

1/2 = 2/4 = 3/6

The fraction wall up to sixths. The red dashed line shows that 1⁄2, 2⁄4 and 3⁄6 all end at the same place.

3/6 = (3 ÷ 3)/(6 ÷ 3) = 1/2

Q4 Are 2⁄3 and 4⁄6 equivalent fractions? Why?

Yes, they are equivalent.

2/3 = (2 × 2)/(3 × 2) = 4/6

On the fraction wall, two strips of 1⁄3 cover exactly the same length as four strips of 1⁄6, because
each third is made of two sixths.

Q5 How many pieces of length 1⁄6 will make a length of 1⁄2?

1/2 = 3/6 = 1/6 + 1/6 + 1/6

Page 25 of 50

Class 6 Maths Chapter 7 Fractions AglaSem · NCERT Solutions

Answer: 3 pieces.

Q6 How many pieces of length 1⁄6 will make a length of 1⁄3?

1/3 = 2/6 = 1/6 + 1/6

Answer: 2 pieces.

Tip: a sixth is half of a third, so it always takes two sixths to make one third.

Figure it Out — Page 165
Section 7.6 Equivalent Fractions

Q1 Are 3⁄6, 4⁄8, 5⁄10 equivalent fractions? Why?

Yes — all three are just other names for one half.

3/6 = (3 ÷ 3)/(6 ÷ 3) = 1/2

4/8 = (4 ÷ 4)/(8 ÷ 4) = 1/2
5/10 = (5 ÷ 5)/(10 ÷ 5) = 1/2

On the fraction wall their shaded lengths end at exactly the same line, so they denote the same
length in different fractional units.

The pattern: in each one the numerator is exactly half the denominator, so half the
strip is shaded.

Page 26 of 50

Class 6 Maths Chapter 7 Fractions AglaSem · NCERT Solutions

Q2 Write two equivalent fractions for 2⁄6.

2/6 = (2 ÷ 2)/(6 ÷ 2) = 1/3 (divide by 2)

2/6 = (2 × 3)/(6 × 3) = 6/18 (multiply by 3)

Also 2/6 = 3/9 = 4/12 = 5/15 …

Answer: 1⁄3 and 3⁄9 are two easy ones (any pair from the list is correct).

4⁄
Q3 6 = ___ = ___ = ___ = … (Write as many as you can)

Divide by 2 once, then keep multiplying the simplest form by 2, 3, 4, …

4/6 = 2/3 = 6/9 = 8/12 = 10/15 = 12/18 = 14/21 = 20/30 = …

The list never ends, because you can multiply the numerator and denominator of 2⁄3 by any
number you like.

Tip: 2⁄3 is the simplest form — the smallest pair of numbers in the whole family.

Figure it Out — Page 166

Page 27 of 50

Class 6 Maths Chapter 7 Fractions AglaSem · NCERT Solutions

Understanding equivalent fractions using equal shares

Q1 Three rotis are shared equally by four children. Show the division in the picture and
write a fraction for how much each child gets. Also, write the corresponding
division facts, addition facts, and multiplication facts.

Three rotis and four children, as given in the book.

Cut each of the 3 rotis into 4 equal parts. That gives 12 quarters, and 12 ÷ 4 = 3 quarters for
every child.

Fraction of roti each child gets = 3/4

Division fact: 3 ÷ 4 = 3/4

Addition fact: 3 = 3/4 + 3/4 + 3/4 + 3/4

Multiplication fact: 3 = 4 × 3/4

Check it yourself: 3/4 + 3/4 + 3/4 + 3/4 = 12/4 = 3 rotis ✔

Page 28 of 50

as e
Class 6 Maths Chapter 7 Fractions
a g l AglaSem · NCERT Solutions

co m
m.
Draw a picture to show how much each child gets when 2 rotis are shared equally
e
Q2

com facts. l as
by 4 children. Also, write the corresponding division facts, addition facts, and
g
. a
em
multiplication

a s
a gl

m
Cut both rotis into halves. That makes 4 halves — one half for each child.
. co ag
e m
g l as
a
Each child gets 2/4 = 1/2 roti

Division fact: 2 ÷ 4 = 2/4 = 1/2

co m
m.
Addition fact: 2 = 1/2 + 1/2 + 1/2 + 1/2

m as e
.co
Multiplication fact: 2 = 4 × 1/2
a g l
se m
g l a
a
m a s
.co agl
Q3 Anil was in a group where 2 cakes were divided equally among 5 children. How

m
much cake would Anil get?

l a se
ANSWER a g
Cut each cake into 5 equal parts: 2 × 5 = 10 pieces of 1⁄5 cake, shared by 5 children.
co m
m .
m as e
.co a g l
m
2 ÷ 5 = 2/5 cake

a s eEach
gl
child: 1/5 + 1/5 = 2/5
a
se m
Answer: Anil gets 2⁄5 of a cake.
com g l a
m . a
ase
agl
In-text Questions — Pages 166 – 168
co m
Equal shares and equivalent fractions
m .
o m l a se
m .c Now, if there are 10 children in my group, how manyagcakes will I need so that they
a se
Q1

agl
get same amount of cake as Anil?

.c
s e m
m a
o agl

. c
Anil’s share is 2⁄
se
5 cake. We need 10 mchildren each to get 2⁄5.
l a
ag

co m
m .
m ase
.co


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Class 6 Maths Chapter 7 Fractions AglaSem · NCERT Solutions

2/5 = (2 × 2)/(5 × 2) = 4/10

So 4 cakes shared by 10 children give 4 ÷ 10 = 4/10 = 2/5 cake each

Answer: 4 cakes are needed. Two groups of “2 cakes for 5 children” put together make exactly
“4 cakes for 10 children”, which is why 2⁄5 = 4⁄10.

Q2 Let us examine the shares of each child in the following situations: 1 roti divided
equally between 2 children; 2 rotis divided equally among 4 children; 3 rotis divided
equally among 6 children. Did you notice that in each situation the share of every
child is the same?

Yes — every child gets half a roti in all three situations.

1 ÷ 2 = 1/2

2 ÷ 4 = 2/4

3 ÷ 6 = 3/6

and 1/2 = 2/4 = 3/6

Why: in each case the number of rotis and the number of children grow in the same
proportion — twice the rotis for twice the children, three times the rotis for three
times the children. So the share per child never changes.

Q3 Find some more fractions equivalent to 1⁄2. Write them in the boxes here.

Multiply the numerator and denominator of 1⁄2 by the same number.

1/2 = 4/8 = 5/10 = 6/12 = 10/20 = 25/50 = 50/100 …

In every one of them the denominator is exactly twice the numerator.

Page 30 of 50

Class 6 Maths Chapter 7 Fractions AglaSem · NCERT Solutions

Q4 Equally divide the rotis in the situations shown below and write down the share of
each child. Are the shares in each of these cases the same? Why? (2 rotis divided
equally among 3 children; 4 rotis divided equally among 6 children; 6 rotis divided
equally among 9 children.) Do you notice anything about the relationship between
the numerator and denominator in each of these fractions?

2 rotis divided equally among 3 children 4 rotis divided equally among 6 children

2 2 2
3 3 3
6 rotis divided equally among 9 children

The three situations given in the book.

2 rotis ÷ 3 children = 2/3 roti each

4 rotis ÷ 6 children = 4/6 roti each

6 rotis ÷ 9 children = 6/9 roti each

Yes, all three shares are the same: 2⁄3 = 4⁄6 = 6⁄9.

Relationship between the numbers:

Page 31 of 50

Class 6 Maths Chapter 7 Fractions AglaSem · NCERT Solutions

4/6 = (2 × 2)/(3 × 2) 6/9 = (2 × 3)/(3 × 3)

Backwards: 4/6 = (4 ÷ 2)/(6 ÷ 2) = 2/3, 6/9 = (6 ÷ 3)/(9 ÷ 3) = 2/3

In each fraction the denominator is one-and-a-half times the numerator, and dividing both by
their common factor always gives 2⁄3 — the simplest form of 4⁄6 and of 6⁄9.

Figure it Out — Pages 168 & 169
Section 7.6 Equivalent Fractions

MATH TALK

Q1 Find the missing numbers: a. 5 glasses of juice shared equally among 4 friends is
the same as ____ glasses of juice shared equally among 8 friends. So, 5⁄4 = ?⁄8. b. 4 kg
of potatoes divided equally in 3 bags is the same as 12 kgs of potatoes divided
equally in ___ bags. So, 4⁄3 = 12⁄?. c. 7 rotis divided among 5 children is the same as ____

rotis divided among ____ children. So, 7⁄5 = ___.

(a) The number of friends is doubled (4 → 8), so the juice must be doubled too.

5/4 = (5 × 2)/(4 × 2) = 10/8 → 10 glasses

(b) The potatoes are tripled (4 kg → 12 kg), so the number of bags must be tripled.

4/3 = (4 × 3)/(3 × 3) = 12/9 → 9 bags

(c) Any equivalent fraction of 7⁄5 works. Doubling both numbers:

7/5 = 14/10 → 14 rotis divided among 10 children

(also 21 rotis among 15 children, 28 rotis among 20 children, …)

Page 32 of 50

Class 6 Maths Chapter 7 Fractions AglaSem · NCERT Solutions

Why the share does not change: multiply the number of things and the number of
people by the same number and every person still receives exactly as much as
before.

In-text Questions — Pages 169 & 170
Comparing shares using equivalent fractions

MATH TALK

Q1 In which group will each child get more chikki? 1 chikki divided between 2 children
or 5 chikkis divided among 8 children.

Compare 1⁄2 and 5⁄8. Make the fractional units the same — eighths.

1/2 = (1 × 4)/(2 × 4) = 4/8

4/8 < 5/8, so 1/2 < 5/8

Answer: the children of the second group (5 chikkis among 8) get more chikki each.

Q2 What about the following groups? In which group will each child get more? 1 chikki
divided between 2 children or 4 chikkis divided among 7 children.

Compare 1⁄2 and 4⁄7. Write 1⁄2 with numerator 4:

1/2 = (1 × 4)/(2 × 4) = 4/8

4 chikkis among 7 children → 4/7 each

4 chikkis among 8 children → 4/8 each

Fewer children share the same 4 chikkis, so 4/7 > 4/8 = 1/2

Answer: the second group (4 chikkis among 7 children) gets more.

Page 33 of 50

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Class 6 Maths Chapter 7 Fractions
a g l AglaSem · NCERT Solutions

co m
m.
Note: the printed line “We must compare 1⁄7 and 4⁄7” is a misprint — the two shares

m as e
.co
being compared are 1⁄2 and 4⁄7, which is exactly what the working below it does.
a g l
se m
g l a
a

com
Q3 Suppose the number of children is kept the same, but the number of units that are

.
m1 2 3
being shared is increased? What can you say about each child’s share now? Why?
e ag
Discuss how your reasoning explains ⁄5 < ⁄5,
g l as ⁄7 < 4⁄7, and 1⁄2 < 5⁄8.
a
m

co
m.
Each child’s share becomes larger. The same children now have more to share, so everybody

m as e
.co l
gets more.
a g
s m⁄5 — 5 children share 1 roti, then 2 rotis. Same children, more rotis → bigger share.
5e<
a
1⁄ 2

l
ag 3⁄7 < 4⁄7 — 7 children share 3 rotis, then 4 rotis. Same fractional unit 1⁄7, more of them.
m a s
1⁄ 5⁄ 1⁄ 4⁄ . Now both use eighths, and 4 eighths < 5 eighths.
2< 8 — first write 2=

.co agl
8

se m
g l a
The rule in one line: with the same denominator, the fraction with the bigger
a
numerator is bigger — you simply have more of the same-sized pieces.

co m
m .
m as e
.co a g l
Now, decide in which of the two groups will each child get a larger share: 1. Group 1:
m 3 glasses of sugarcane juice divided equally among 4 children. Group 2: 7 glasses of
Q4

l a se
ag sugarcane juice divided equally among 10 children. 2. Group 1: 4 glasses of

se m
sugarcane juice divided equally among 7 children. Group 2: 5 glasses of sugarcane

com
juice divided equally among 7 children. Which groups were easier to compare? Why?
g l a
m . a
gl ase
a

Pair 1 — compare 3⁄ and 7⁄ . The denominators are different, so make them the same. A
4 10

co m
.
common multiple of 4 and 10 is 20 (their product 40 also works).

e m
com= (3 × 5)/(4 × 5) = 15/20 g l as
m .3/4 a
l a se
ag 7/10 = (7 × 2)/(10 × 2) = 14/20
.c
15/20 > 14/20, so 3/4 > 7/10
s e m
com a
.
em share. agl
a s
ag5l⁄7. The fractional unit is already the same.
→ Group 1 children get the larger

Pair 2 — compare 4⁄7 and

co m
m .
m as e
.co


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Class 6 Maths Chapter 7 Fractions AglaSem · NCERT Solutions

5/7 > 4/7 → Group 2 children get the larger share.

The second pair was much easier to compare, because the number of children is the same in
both groups — the fractions already have the same denominator, so we only had to compare
the numerators.

In-text Question — Page 172
Equivalent fractions with the same fractional unit

Q1 Find equivalent fractions for the given pairs of fractions such that the fractional
units are the same. a. 7⁄2 and 3⁄5 b. 8⁄3 and 5⁄6 c. 3⁄4 and 3⁄5 d. 6⁄7 and 8⁄5 e. 9⁄4 and 5⁄2 f.
1⁄ 2⁄ 8⁄ 11⁄ 13⁄ 1⁄
10 and 9 g. 3 and 4 h. 6 and 9

Take a common multiple of the two denominators (the smallest one is easiest) and rewrite both
fractions with it.

PAIR COMMON DENOMINATOR EQUIVALENT FRACTIONS

a. 7/2, 3/5 10 35/10 and 6/10

b. 8/3, 5/6 6 16/6 and 5/6

c. 3/4, 3/5 20 15/20 and 12/20

d. 6/7, 8/5 35 30/35 and 56/35

e. 9/4, 5/2 4 9/4 and 10/4

f. 1/10, 2/9 90 9/90 and 20/90

g. 8/3, 11/4 12 32/12 and 33/12

h. 13/6, 1/9 18 39/18 and 2/18

Sample working for (d):

6/7 = (6 × 5)/(7 × 5) = 30/35

8/5 = (8 × 7)/(5 × 7) = 56/35

Page 35 of 50

Class 6 Maths Chapter 7 Fractions AglaSem · NCERT Solutions

Tip: when one denominator is already a multiple of the other — as in (b) 3 and 6, or
(e) 2 and 4 — only one fraction needs to be changed.

Figure it Out — Page 173
Expressing a fraction in lowest terms

Q1 Express the following fractions in lowest terms: a. 17⁄51 b. 64⁄144 c. 126⁄147 d. 525⁄112

Divide the numerator and the denominator by their highest common factor.

a. 51 = 17 × 3, so the HCF is 17
17/51 = (17 ÷ 17)/(51 ÷ 17) = 1/3

b. 64 = 16 × 4, 144 = 16 × 9, so the HCF is 16

64/144 = (64 ÷ 16)/(144 ÷ 16) = 4/9

In steps: 64/144 = 32/72 = 16/36 = 8/18 = 4/9

c. 126 = 21 × 6, 147 = 21 × 7, so the HCF is 21

126/147 = (126 ÷ 21)/(147 ÷ 21) = 6/7

d. 525 = 7 × 75, 112 = 7 × 16, so the HCF is 7

525/112 = (525 ÷ 7)/(112 ÷ 7) = 75/16 (= 4 11/16)

How to be sure it is lowest: check that the two final numbers share no factor except
1 — 4 and 9, 6 and 7, 75 and 16 have none.

Figure it Out — Page 174

Page 36 of 50

Class 6 Maths Chapter 7 Fractions AglaSem · NCERT Solutions

Section 7.7 Comparing Fractions

Q1 Compare the following fractions and justify your answers: a. 8⁄3, 5⁄2 b. 4⁄9, 3⁄7 c. 7⁄10,
9⁄ 12⁄ , 8⁄ e. 9⁄ , 5⁄
14 d. 5 5 4 2

Make the fractional units the same, then compare the numerators.

a. common denominator 6: 8/3 = 16/6, 5/2 = 15/6

16/6 > 15/6 → 8/3 > 5/2

b. common denominator 63: 4/9 = 28/63, 3/7 = 27/63

28/63 > 27/63 → 4/9 > 3/7

c. common denominator 70: 7/10 = 49/70, 9/14 = 45/70

49/70 > 45/70 → 7/10 > 9/14

d. same fractional unit already: 12 fifths > 8 fifths

→ 12/5 > 8/5

e. common denominator 4: 9/4 stays, 5/2 = 10/4

9/4 < 10/4 → 9/4 < 5/2

Q2 Write the following fractions in ascending order. a. 7⁄10, 11⁄15, 2⁄5 b. 19⁄24, 5⁄6, 7⁄12

(a) The smallest common multiple of 10, 15 and 5 is 30.

7/10 = 21/30 11/15 = 22/30 2/5 = 12/30

12/30 < 21/30 < 22/30

Page 37 of 50

Class 6 Maths Chapter 7 Fractions AglaSem · NCERT Solutions

Ascending order: 2⁄5 < 7⁄10 < 11⁄15

(b) The smallest common multiple of 24, 6 and 12 is 24.

19/24 stays 5/6 = 20/24 7/12 = 14/24
14/24 < 19/24 < 20/24

Ascending order: 7⁄12 < 19⁄24 < 5⁄6

Q3 Write the following fractions in descending order. a. 25⁄16, 7⁄8, 13⁄4, 17⁄32 b. 3⁄4, 12⁄5, 7⁄12,
5⁄
4

(a) The smallest common multiple of 16, 8, 4 and 32 is 32.

25/16 = 50/32 7/8 = 28/32 13/4 = 104/32 17/32 stays

104/32 > 50/32 > 28/32 > 17/32

Descending order: 13⁄4 > 25⁄16 > 7⁄8 > 17⁄32

(b) The smallest common multiple of 4, 5 and 12 is 60.

3/4 = 45/60 12/5 = 144/60 7/12 = 35/60 5/4 = 75/60

144/60 > 75/60 > 45/60 > 35/60

Descending order: 12⁄5 > 5⁄4 > 3⁄4 > 7⁄12

Quick check: 12⁄5 and 5⁄4 are more than 1, while 3⁄4 and 7⁄12 are less than 1 — so the
first two must come first.

In-text Questions — Pages 175 & 177

Page 38 of 50

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Class 6 Maths Chapter 7 Fractions
a g l AglaSem · NCERT Solutions

Section 7.8 Addition and Subtraction of Fractions
co m
e m.
m l as
.co a g
Meena ate 1⁄2 of the chikki and her younger brother ate 1⁄4 of it. How much of the
emchikki did Meena and her brother eat together?
Q1

a s
gl
total
a
m

. co ag
m
Measure both shares with the same fractional unit, 1⁄4. Meena’s half is 2 quarters.

as e
a g l
Total eaten = 1/2 + 1/4

co m
m.
= 1/4 + 1/4 + 1/4

m as e
.co
= 3 × 1/4 = 3/4 of the chikki
a g l
a s em
a l
gAnswer: together they ate 3⁄4 of the chikki.

m a s
m .co agl
l a se
How much of the total chikki is remaining?
g
Q2

a
m

. co
The whole chikki is 4 quarters and 3 quarters have been eaten.
e m
m l as
m .co a g
l a se 1 − 3/4 = 4/4 − 3/4 = 1/4

ag
Answer: 1⁄4 of the chikki is left.
se m
com g l a
m . a
ase
Q3
agl
Try adding 4⁄7 + 6⁄7 using a number line. Do you get the same answer?

co m
m .
as e

m l
.co a g
Yes — the number line gives the same answer. Mark the unit in sevenths, jump 4 sevenths
mfrom 0, then 6 sevenths more.
l a se
ag
.c
s e m
m a
e m . co agl
g l as
a

co m
m .
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.co


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Class 6 Maths Chapter 7 Fractions AglaSem · NCERT Solutions

10/7
4/7 6/7

0 1 2

Two jumps on a number line marked in sevenths: 4 sevenths, then 6 sevenths more, landing on 10⁄7
— a little past 1.

4/7 + 6/7 = 10/7

= 7/7 + 3/7 = 1 + 3/7 = 1 3/7

Why the denominators stay 7: both fractions are counted in the same fractional
unit 1⁄7. Adding 4 sevenths and 6 sevenths simply gives 10 sevenths, just as 4
mangoes and 6 mangoes give 10 mangoes.

Figure it Out — Page 179
Brahmagupta’s method for adding fractions

Q1 Add the following fractions using Brahmagupta’s method: a. 2⁄7 + 5⁄7 + 6⁄7 b. 3⁄4 + 1⁄3 c.
2⁄ 5⁄ 2⁄ 2⁄ 3⁄ 1⁄ 1⁄ 2⁄ 4⁄ 4⁄ 2⁄ 3⁄ 5⁄ 9⁄ 5⁄ 8⁄ 2⁄ 3⁄
3+ 6 d. 3+ 7 e. 4+ 3+ 5 f. 3+ 5 g. 5+ 3 h. 5+ 8 i. 2+ 4 j. 3+ 7 k. 4+
1⁄ 1⁄ 2⁄ 4⁄ 3⁄ 9⁄ 5⁄ 7⁄
3+ 5 l. 3+ 5+ 7 m. 2+ 4+ 6

Brahmagupta’s three steps every time: (1) make the fractional units the same, (2) add the
numerators, (3) write the answer in lowest terms.

Page 40 of 50

Class 6 Maths Chapter 7 Fractions AglaSem · NCERT Solutions

QUESTION WITH A COMMON FRACTIONAL UNIT ANSWER

a. 2/7 + 5/7 + 6/7 (2 + 5 + 6)/7 13/7 = 1 6/7

b. 3/4 + 1/3 9/12 + 4/12 13/12 = 1 1/12

c. 2/3 + 5/6 4/6 + 5/6 = 9/6 3/2 = 1 1/2

d. 2/3 + 2/7 14/21 + 6/21 20/21

e. 3/4 + 1/3 + 1/5 45/60 + 20/60 + 12/60 77/60 = 1 17/60

f. 2/3 + 4/5 10/15 + 12/15 22/15 = 1 7/15

g. 4/5 + 2/3 12/15 + 10/15 22/15 = 1 7/15

h. 3/5 + 5/8 24/40 + 25/40 49/40 = 1 9/40

i. 9/2 + 5/4 18/4 + 5/4 23/4 = 5 3/4

j. 8/3 + 2/7 56/21 + 6/21 62/21 = 2 20/21

k. 3/4 + 1/3 + 1/5 45/60 + 20/60 + 12/60 77/60 = 1 17/60

l. 2/3 + 4/5 + 3/7 70/105 + 84/105 + 45/105 199/105 = 1 94/105

m. 9/2 + 5/4 + 7/6 54/12 + 15/12 + 14/12 83/12 = 6 11/12

Sample working for (c):

The smallest common multiple of 3 and 6 is 6

2/3 = (2 × 2)/(3 × 2) = 4/6, 5/6 stays 5/6

4/6 + 5/6 = 9/6 = (9 ÷ 3)/(6 ÷ 3) = 3/2 = 1 1/2

Did you notice? (f) and (g) give the same answer, and so do (e) and (k) — the order
of the fractions does not change the sum.

Page 41 of 50

Class 6 Maths Chapter 7 Fractions AglaSem · NCERT Solutions

Q2 Rahim mixes 2⁄3 litres of yellow paint with 3⁄4 litres of blue paint to make green
paint. What is the volume of green paint he has made?

Volume = 2/3 + 3/4

Common denominator = 3 × 4 = 12

2/3 = 8/12, 3/4 = 9/12

8/12 + 9/12 = 17/12 = 1 5/12 litres

Answer: Rahim has made 17⁄12 litres, that is 15⁄12 litres of green paint.

Check it yourself: 17⁄12 is a little more than 1 litre — sensible, since 2⁄3 and 3⁄4 are
each a bit less than a litre.

Q3 Geeta bought 2⁄5 meter of lace and Shamim bought 3⁄4 meter of the same lace to put
a complete border on a table cloth whose perimeter is 1 meter long. Find the total
length of the lace they both have bought. Will the lace be sufficient to cover the
whole border?

Total lace = 2/5 + 3/4

Common denominator = 5 × 4 = 20

2/5 = 8/20, 3/4 = 15/20
8/20 + 15/20 = 23/20 m = 1 3/20 m

The border needs 1 m and they have 23⁄20 m, which is more than 1 m (since 20⁄20 = 1).

Answer: total lace = 13⁄20 m. Yes, it is sufficient — 3⁄20 m of lace is left over.

In-text Question — Page 180

Page 42 of 50

Class 6 Maths Chapter 7 Fractions AglaSem · NCERT Solutions

Subtracting fractions with the same fractional unit

Q1 So, we are left with 2 shaded parts, i.e., 6⁄7 – 4⁄7 = 2⁄7. Try doing this same exercise
using the number line.

6
7

Fractional parts to be removed.

The rectangular strip model: 6/7 shaded, then 4 shaded parts taken away.

Mark the unit in sevenths. Go forward to 6⁄7, then jump back 4 sevenths.

Start at 6/7

Jump back 4 steps of 1/7: 6/7 → 5/7 → 4/7 → 3/7 → 2/7

So 6/7 − 4/7 = 2/7

The number line gives the same answer as the strip picture.

Why the denominator does not change: subtraction is “take away”. Taking 4 pieces
of size 1⁄7 from 6 pieces of the same size leaves 2 pieces of size 1⁄7.

Figure it Out — Page 181

Page 43 of 50

as e
Class 6 Maths Chapter 7 Fractions
a g l AglaSem · NCERT Solutions

Subtraction with the same denominator
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5⁄ 3

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Answer: 2⁄8, which in lowest terms is 1⁄4.

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7 5
Q2

m a s
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se m
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a

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Figure it Out — Page 182 s e m
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m ase
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a g l Page 44 of 50

Class 6 Maths Chapter 7 Fractions AglaSem · NCERT Solutions

Brahmagupta’s method for subtracting fractions

Q1 Carry out the following subtractions using Brahmagupta’s method: a. 8⁄15 – 3⁄15 b. 2⁄5
– 4⁄15 c. 5⁄6 – 4⁄9 d. 2⁄3 – 1⁄2

a. Same fractional unit already:

8/15 − 3/15 = 5/15 = (5 ÷ 5)/(15 ÷ 5) = 1/3

b. 15 is a multiple of 5, so change only the first fraction:

2/5 = (2 × 3)/(5 × 3) = 6/15

6/15 − 4/15 = 2/15

c. The smallest common multiple of 6 and 9 is 18:

5/6 = 15/18, 4/9 = 8/18

15/18 − 8/18 = 7/18

d. Common denominator 6:

2/3 = 4/6, 1/2 = 3/6

4/6 − 3/6 = 1/6

Q2 Subtract as indicated: a. 13⁄4 from 10⁄3 b. 18⁄5 from 23⁄3 c. 29⁄7 from 45⁄7

“A from B” means B − A.

a. 10/3 − 13/4, common denominator 12:

10/3 = 40/12, 13/4 = 39/12

40/12 − 39/12 = 1/12

Page 45 of 50

Class 6 Maths Chapter 7 Fractions AglaSem · NCERT Solutions

b. 23/3 − 18/5, common denominator 15:

23/3 = 115/15, 18/5 = 54/15

115/15 − 54/15 = 61/15 = 4 1/15

c. 45/7 − 29/7, same fractional unit:

(45 − 29)/7 = 16/7 = 2 2/7

Q3 Solve the following problems: a. Jaya’s school is 7⁄10 km from her home. She takes an
auto for 1⁄2 km from her home daily, and then walks the remaining distance to
reach her school. How much does she walk daily to reach the school? b. Jeevika
takes 10⁄3 minutes to take a complete round of the park and her friend Namit takes
13⁄
4 minutes to do the same. Who takes less time and by how much?

(a) Walking distance = total distance − auto distance.

7/10 − 1/2

1/2 = 5/10

7/10 − 5/10 = 2/10 = 1/5 km

Jaya walks 1⁄5 km (200 m) every day.

(b) Compare 10⁄3 and 13⁄4 using the common denominator 12.

Jeevika: 10/3 = 40/12 minutes

Namit: 13/4 = 39/12 minutes

39/12 < 40/12, so Namit is faster

Difference = 40/12 − 39/12 = 1/12 minute

Namit takes less time — by 1⁄12 of a minute (5 seconds).

Page 46 of 50

Class 6 Maths Chapter 7 Fractions AglaSem · NCERT Solutions

Puzzle — Pages 184 & 185
Section 7.9 A Pinch of History · Egyptian fractions

TRY THIS

Q1 Can you find three different fractional units that add up to 1?

Yes — and there is only one such set: 1⁄2 + 1⁄3 + 1⁄6 = 1.

Common denominator 6:

1/2 = 3/6, 1/3 = 2/6, 1/6 = 1/6

3/6 + 2/6 + 1/6 = 6/6 = 1 ✔

1/6

1/2
1/3

1/2 + 1/3 + 1/6 = 1
One roti cut into three unequal but familiar pieces — a half, a third and a sixth.

Page 47 of 50

Class 6 Maths Chapter 7 Fractions AglaSem · NCERT Solutions

Why nothing else works: start from 1⁄3 + 1⁄3 + 1⁄3 = 1. To make the units different,
one of them must be made bigger — and the only fractional unit bigger than 1⁄3 is
1⁄ . Now 1⁄ + 1⁄ + 1⁄ = 1, and the only way to enlarge a 1⁄ to something new is 1⁄ .
2 2 4 4 4 3

The third piece is then forced: 1 − 1⁄2 − 1⁄3 = 1⁄6.

Q2 Can you find four different fractional units that add up to 1?

Yes. This puzzle has exactly six solutions (the order of the four fractions does not matter).

# FOUR DIFFERENT FRACTIONAL UNITS CHECK WITH A COMMON DENOMINATOR

1 1/2 + 1/3 + 1/7 + 1/42 21/42 + 14/42 + 6/42 + 1/42 = 1

2 1/2 + 1/3 + 1/8 + 1/24 12/24 + 8/24 + 3/24 + 1/24 = 1

3 1/2 + 1/3 + 1/9 + 1/18 9/18 + 6/18 + 2/18 + 1/18 = 1

4 1/2 + 1/3 + 1/10 + 1/15 15/30 + 10/30 + 3/30 + 2/30 = 1

5 1/2 + 1/4 + 1/5 + 1/20 10/20 + 5/20 + 4/20 + 1/20 = 1

6 1/2 + 1/4 + 1/6 + 1/12 6/12 + 3/12 + 2/12 + 1/12 = 1

The easiest one to find is 1⁄2 + 1⁄4 + 1⁄6 + 1⁄12 = 1 — split the 1⁄3 of the three-piece answer into 1⁄4 +
1⁄
12.

Did you know? Writing a number as a sum of different fractional units is called an
Egyptian fraction. The chapter shows another one: 19⁄24 = 1⁄2 + 1⁄6 + 1⁄8. Check it:
12/24 + 4/24 + 3/24 = 19/24 ✔

How to hunt for them: the biggest unit must be 1⁄2 (four units each 1⁄3 or smaller
cannot reach 1 once they are all different). Then find three different units adding to
1⁄ , and repeat the same reasoning.
2

Page 48 of 50

as e
Class 6 Maths Chapter 7 Fractions
a g l AglaSem · NCERT Solutions

co m
se m.
Chapter at amglance
l a
.co ag 1 1 1
a s em
When one whole is divided into equal parts, each part is a fractional unit — ⁄2, ⁄3, ⁄4, …

a l more people share the same thing, the smaller each share: 1⁄9 < 1⁄5.
gThe
A fraction like 3⁄4 is best read as “3 times 1⁄4” — it tells you the fractional unit (1⁄4) and how

co m
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many of them there are (3). The top number is the numerator, the bottom the
ag
denominator.
l a se
agnumber line. If the numerator is smaller than the
Every fraction has a point on the
denominator the fraction is less than 1; if it is bigger, the fraction is more than 1 and can be
written as a mixed number, e.g. 10⁄7 = 13⁄7.
. c om
Equivalent fractions show the same share in different fractional units: 1e⁄2m= 2⁄4 = 3⁄6 = 4⁄8.
c o m (or dividing) the numerator and denominator by the same
g l asnumber never
m . a
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Multiplying
s
gla
changes the value.
a A fraction is in lowest terms (simplest form) when the numerator and denominator have no
common factor except 1. Divide both by their highest common factor: 16⁄20 = 4⁄5.
m a s
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To compare, add or subtract fractions, first make the fractional units the same (a common agl
a s
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denominator) and then compare, add or subtract the numerators. This is Brahmagupta’s
method, written down in 628

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m a s e
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co m
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m ase
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a g l Page 49 of 50

Class 6 Maths Chapter 7 Fractions AglaSem · NCERT Solutions

Quick revision

IDEA RULE IN SHORT EXAMPLE FROM THE RESULT
CHAPTER

Fractional unit 1 whole shared equally by n 1 roti shared by 4 children 1/4 roti
each

Reading a fraction 3/4 = 3 times 1/4 5/6: 5 is numerator, 6 is ‘5 upon 6’
denominator

Fraction on a number 0 to 1 cut into n equal parts unit cut into 5 parts, 4 steps 4/5
line

Fraction greater than 1 numerator > denominator 4/7 + 6/7 = 10/7 1 3/7

Mixed number to (whole × denominator) + numerator 3 1/4 = (3 × 4 + 1)/4 13/4
fraction

Equivalent fractions × or ÷ numerator and denominator by 1/2 = 2/4 = 3/6 = 4/8 same
the same number length

Simplest form divide both by the highest common 16/20 ÷ 4 4/5
factor

Comparing fractions make the denominators the same 4/5 = 36/45, 7/9 = 35/45 4/5 > 7/9

Adding like fractions add numerators, keep the denominator 2/5 + 1/5 3/5

Brahmagupta — common denominator, then add 1/4 + 1/3 = 3/12 + 4/12 7/12
addition

Brahmagupta — common denominator, then subtract 3/4 − 2/3 = 9/12 − 8/12 1/12
subtraction

Egyptian-fraction 1 as a sum of different fractional units 1/2 + 1/3 + 1/6 =1
puzzle

Page 50 of 50