NCERT Solutions Class 3 Maths Chapter 7 Raksha Bandhan

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 3 · M AT H S

NCERT Solutions + ×
Chapter 7: Raksha
Bandhan

NCERT Textbook — Maths Mela

BOOK PAGES SECTIONS QUESTIONS MEDIUM

82 – 106 27 95 English

Every question Step by step Exam ready
Activities, in-text questions The working shown, not Written to match how
and all exercises. just the final answer. marks are awarded.

Solutions, notes, sample papers & more at 87 pages

l h h e k h dhan A aS m · NCERT So ut ons
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C L A SS 3 · M AT H S · M AT H S M E L A
NCERT So ut ons — C apt r 7: Ra s a Ban an
l i h e k h dh

Raksha Bandhan brings Gopal, Dhara and their Atya together. As they make rakhis, buy laddoos and share
sweets, we learn multiplication as repeated addition and division as equal sharing.

T E XT B OOK B OOK PAGE S
Ma h Mela Cla
t s ss 3 82  106

S EC T I ONS QU E ST I ONS
27 95

M EDI UM
English

In-text Questions — Page 82

Page 1 of 87

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Chapter opener  Gopalʼs house

Q1 Carefully observe Gopal s house What do you find interesting here
ʼ . ?

The toran of leaves over the door

Glasses on the table Glasses on the tray

Baskets of pomegranates

Two plates of flowers
Gopa s ous rawn a a n w t t
lʼ h e d g i , i h he le aves, glasses, baskets and flower plates.

Sample answer: The house is ready for the festival. Here are three things I found interesting.
 A green toran of leaves hangs over the door.
 The same things are kept in equal groups  5 pomegranates in every basket.
 The glasses stand in neat rows, so they are easy to count.

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Why Things kept in equal groups are quick to count We count one group then
: . ,

skip count.

Try this: Write two more things you see. Every child may find something
different.

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Find and count the number of each of these objects and write Leaves .com
Glasses
m
Q2
se
. ______

m Flowers
a
agl
Pomegranate
.co
______ ______ ______

se m
g l a
a

co m
em . ag
g l as
a
The toran of leaves over the door
co m
e m.
m l as
m .co a g
l a se
a g
m a s
Glasses on thesteam .co Glasses on the tray agl
gla
ble
a

co m
m .
m as e
.co eg a a e a g l
m Baskets of pom
ase
r n t s

agl
se m
com g l a
m . a
e
as T
agl
wo p lates of flowers
Gopa s ous rawn a a n w t t
lʼ h e d g i , i h he le aves, glasses, baskets and flower plates.

co m
m .
as e
.c omE R
A NSW
a g l
m
se Count each set in groups Do not count one by one
a
agl
. .

.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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OBJECT HOW TO COUNT NUMBER

Leaves 12 on one side, 12 on the other 24

Glasses 20 on the table and 10 on the tray 30

Pomegranate 4 baskets, 5 in each basket 20

Flowers 2 plates, 10 in each plate 20

− Exp anat on
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Why Counting in equal groups is faster baskets of is
: . 4 5 5  5  5  5  20.

Tip: The small basket near the stool also has a few loose flowers. The count 20
is for the flowers on the two plates.

C elebration Begins! / Letʼs Make Rakhis — Page 83
Guessing the festival and planning the rakhis

Q1 Dhara s mother has bought some material for the festival Guess the festival they are
ʼ .

preparing for.

The picture on page 83
Dhara s mother stands behind a round green table in her room On the
ʼ .

table are a tray of bright flowers in many colours, a glass box full of
coloured beads, an open box of sweets, coloured threads lying loose,
and a gift wrapped in yellow paper with a purple ribbon. A window and
two framed pictures are on the wall behind her. Dhara stands at the right
and her speech bubble says, “Raksha bandhan!ˮ

The festival material on the table; described here, see the textbook for the painting.

They are getting ready for Raksha Bandhan.

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On the table are coloured threads, beads and flowers. These are used to make rakhis.
There are sweets and a gift too.
Dhara herself says, “Raksha bandhan!ˮ

− Exp anat on
l i

Why We tie rakhis on Raksha Bandhan So things for making rakhis tell us the
: .

festival.

Q2 For each Rakhi we need one flower
, , .......... t hreads and .......... beads.

One rakhi
One rakhi with its flower, threads and beads.

Look at one rakhi in the picture and count its parts.

For each rakhi we need

1 flower

2 threads

4 beads

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Why The thread goes out on both sides of the flower Each side has beads so
: . 2 , 2

+ 2  4 beads.

Q3 We need to make 5 Rakhis. How many threads, flowers and beads do they need?

one flower for each rakhi

two threads for each rakhi

four beads for each rakhi
The flowers, the threads and the beads laid out for five rakhis.

One rakhi takes 1 flower, 2 threads and 4 beads. Take that 5 times.

Flowers: 5 × 1 = 5

Threads: 5 × 2 = 10

Beads: 5 × 4 = 20

Why Every rakhi needs the same parts So we add the same number times
: . 5 ,

which is multiplication.

In-text Questions — Page 84

Page 7 of 87

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Making 5 rakhis

Q1 Each Rakhi uses flower We need to make Rakhis
1 . 5 . 1  1  1  1  1  ...... This can
also be said as times or
5 1For Rakhis we need
5  1  5. 5 , ......... flowers.

one flower for each rakhi

two threads for each rakhi

four beads for each rakhi
The flowers, the threads and the beads laid out for five rakhis.

Five rakhis need 5 flowers .

1+1+1+1+1=5

5 times 1

5×1=5

Why Each rakhi takes just flower So rakhis take flowers one for each
: 1 . 5 5 , .

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Each Rakhi uses . comor 5
sem
Q2 hreads. We need to make 5 Rakhis. 2  2  2  2  2  ......
2 t

m
times 2 or 5  2  ...... For 5 Rakhis, we need .............. threads.
a
em . co agl
g l as
a

co m
one flower for each rakhi
em . ag
g l as
a

co m
two threads for each rakhi
e m.
m l as
m .co a g
l a se
a g
four beads for each rakhi
m a s
.co agl
The flowers, the threads and the beads laid out for five rakhis.

se m
g l a
a
co m
Five rakhis need 10 threads
m .
e
.

c o m g l as
.
2m+ 2 + 2 + 2 + 2 = 10 a
a s e
agl 5 times 2

se m
com a
5 × 2 = 10
. ag l
m
ase
− Exp anat on
l i
agl
Why Adding the same number again and again five times i.scothem same as
m
: 2 , , 5 

a se
com g l
2.

m . a
ase
agl
Tip: Skip count by 2  2, 4, 6, 8, 10. You say five numbers and stop at 10.
.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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Q3 Each Rakhi uses beads We need to make
4 . 5 Rakhis. 4  4  4  4  4  ...... or 5 times
4 or For Rakhis we need
5  4  ...... 5 , .............. beads.

one flower for each rakhi

two threads for each rakhi

four beads for each rakhi
The flowers, the threads and the beads laid out for five rakhis.

Five rakhis need 20 beads .

4 + 4 + 4 + 4 + 4 = 20

5 times 4

5 × 4 = 20

Why There are equal groups of beads Five fours make
: 5 4 . 20.

Tip: Skip count by 4  4, 8, 12, 16, 20.

Try it Out! — Page 85

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Rakhis for more people

TRY THIS

Q1 For making 10 such Rakhis, we need ......... flowers, ......... threads and ......... beads.

One rakhi
One rakhi with its flower, threads and beads.

For 10 rakhis, take each part 10 times.

Flowers: 10 × 1 = 10

Threads: 10 × 2 = 20

Beads: 10 × 4 = 40

Why : 10 rakhis are double of 5 rakhis. So each number also becomes double.

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Q2 There are 30 flowers, 30 threads and 30 beads. How many Rakhis can you make with
this material? Use drawings if needed to find out the answer.

One rakhi
One rakhi with its flower, threads and beads.

We can make 7 rakhis.

Each rakhi needs 4 beads.

7 × 4 = 28 beads. 2 beads are left.

8 rakhis would need 32 beads. We have only 30.

Threads 14 and flowers 7 are enough for 7 rakhis.

− Exp anat on
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 30 flowers are enough for 30 rakhis, because 1 flower goes into 1 rakhi.
 30 threads are enough for 15 rakhis, because 2 threads go into 1 rakhi.
 30 beads are enough for only 7 rakhis, because 4 beads go into 1 rakhi.

7 rakhis take 7 flowers, 14 threads and 28 beads.

Left over: 23 flowers, 16 threads, 2 beads.

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Why The beads run out first so the beads decide how many rakhis we can
: ,

make.

Q3 FUN ACTIVITY Try making a colourful Rakhi at your home Show it in class
 . .

Sample answer: I made a rakhi with 1 paper flower, 2 threads and 4 beads. I showed it in class.

Tip: Use a paper or cloth flower, a long thread and a few beads. Count the parts
while you make it.

In-text Questions — Page 86

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Jagannath Sweet Shop
co m
se m.
Q1 com
.
How would we count the laddoos in this box
a ghlraee times three
emals 9 or, 3  3  ...... There are ...... laddoos in this box.
? 3  3  3  ...... or, t

a s
equ
agl

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
m .co agl
l a se
ag

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

m
A box of laddoos as e m . co m

.co a g l
m
ase
The box of laddoos from the sweet shop.

agl c
m .
a s e
. c om agl
Count the laddoos row by row eTm
a s here are rows of

agl
. 3 3.

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m .
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.co

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3+3+3=9

three times three equals 9

3×3=9

There are 9 laddoos in this box.

− Exp anat on
l i

Why Counting
: 1, 2, 3, 4 … gets muddled. Equal rows let us add 3 just three
times.

Q2 Please give me 2 boxes of laddoos. 9  9  18, 2  9  18. Also, 3  3  3  3  3  3 
....... Or, 6 times 3 equals 18. Or, 6  3  18.

Two boxes of laddoos
The two boxes of laddoos Dhara asked for.

Two boxes hold 18 laddoos.

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3 + 3 + 3 + 3 + 3 + 3 = 18

6 times 3 = 18

9 + 9 = 18, so 2 × 9 = 18

− Exp anat on
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Why Two boxes have rows of in all We can add the rows or add the two
: 6 3 . ,

boxes. The laddoos are the same.

In-text Questions — Page 87
Sharing 18 laddoos

Q1 After Atya and children come we will be eople in the house. When we distribute 18
, 9 p

laddoos equally to all, how many will each of us get?

The box of laddoos
The laddoos that are to be shared at home.

Each person gets 2 laddoos .

18 shared equally by 9

18 ÷ 9 = 2 laddoos

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Why Give one laddoo to each of the people
: 9 . 9 are used and 9 are left. Give one
more each, and none is left.

Q2 Imagine yourself to be Dhara Distribute laddoos equally among nine of your friends
. 18 .

Items left Items left
: ........ When laddoos are shared equally among nine
: ........ 18

people, each of them gets ............ laddoos.

The 18 laddoos and the nine empty plates.

Put one laddoo in each plate, then go round again.

ROUND LADDOOS GIVEN ITEMS LEFT

First 1 to each of 9 18 – 9 = 9

Second 1 more to each of 9 9–9=0

Each of them gets 2 laddoos . 18  9  2.

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N n p at s a t r s ar n
i e l e f e h i g

Why Sharing one at a time keeps every plate equal We stop when nothing is left
: . .

Try it Out! — Pages 88–89
Sharing kaju katlis among 5 people

TRY THIS

Q1 Look at the figure carefully. Estimate the number of kaju katlis. Count and write the
number of kaju katlis. Total number of kaju katlis = .........

The kaju katlis lying in the tray.

First guess then count in rows
, .

 My estimate: about 20.

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 There are 4 rows with 5 kaju katlis in each row.
co m
 5  5  5  5  20.
em.
Total number.coof m g l as
m
kaju katlis 20
a
se
= .

l a
a−gExp anat on
l i

Why A guess tells us if our count isms.ecnsomible is close to the guess, so the ag
e
:

as
. 20

count is right.
a g l

co m
se m.
Q2 . coibmute all kaju katlis equally among
Distr eople. You can do ig
tblay drawing kaju katlis on
a
sem
5 p

the plates How many will each get
a
agl
. ?

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
a
se m
com g l a
m . a
e
Tlastray o a u at s an t
a g he f k j k li d he fi ve empty plates.

co m
m .
ase
com
Ea.ch person gets 4 kaju katlis
a g l
em Put in each plate
.

a s
gl
are left.
a
1 . 15

 Put 1 more in each. 10 are left.
.c
s e m
m a
 Put 1 more in each. 5 are left.
 Put 1 more in each. None is left..co
m agl
l a se
a g

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m .
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20 equally shared by 5 is 4 each.

20 ÷ 5 = 4

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Why We could go round four times before the tray became empty
: .

Q3 Compare your work with Dhara s work Items left ʼ . : ........ Items left
: ........ Items left
:

Items left
........ : ........

ROUND TAKEN AWAY ITEMS LEFT

Start 20

1 –5

2 –5

3 –5

4 –5

Dhara takes away 5 each time and writes what is left.

20 – 5 = 15

15 – 5 = 10

10 – 5 = 5

5–5=0

She subtracted 5 four times, so each person got 4.

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− Exp anat on
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Why Each round gives to all people so go away every time Counting the
: 1 5 , 5 .

rounds gives the share.

Let us o — Pages 90–91
D

Sharing sweets equally

Q1 Distribute all the kaju katlis equally among eople. How many kaju katlis will each
4 p

get? Let us do this in the picture given below. Strike out the kaju katlis from the tray
and draw them in the plates. Each will get ........ kaju katlis. 16  4  ........

The tray of kaju katlis and the four empty plates.

Each will get 4 kaju katlis .

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16 – 4 = 12 left

12 – 4 = 8 left

8 – 4 = 4 left
4 – 4 = 0 left

16  4  4.

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Why We took away four times before the tray was empty So each plate got
: 4 .

4.

Q2 Distribute edas in plates equally among 5 people. How many pedas will each get?
15 p

15 equally shared by 5 is ........ each. 15  5  ........ .

15 pedas and five plates

The pedas and the five plates they go into.

Each person gets 3 pedas .

15 – 5 = 10 left

10 – 5 = 5 left

5 – 5 = 0 left

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15 equally shared by 5 is 3 each. 15  5  3.

− Exp anat on
l i

Why The pedas are in rows of One whole row goes round once so there
: 3 5. ,

are 3 rounds.

Let us Think — Pages 91–92
Cycle wheels and jalebis
MATH TALK

Q1 Each cycle needs 2 wheels. How many cycles can be fitted with 12 wheels? 12 equally
divided by 2 is ........ . 12  2  ........ .

Cycle frames with no wheels

The wheels lying ready
The cycle frames and the wheels lying beside them.

6 cycles can be fitted.
 Put the wheels in pairs: 2, 2, 2, 2, 2, 2.
 Count the pairs — there are 6.

12 equally divided by 2 is 6.

12 ÷ 2 = 6

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co m
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o m l a se
ag ber of cycles
WheymE.vcery cycle takes one pair The number of pairs is the num
as
: . .

a g l
The picture has 8 frames, so 2 frames will still have no wheels.
co m
em . ag
g l as
a
Q2 Look at the picture carefully. Count the number of jalebis. There are ........ jalebis. How
co m
m.
did you count? Discuss with your friends.
m as e
.co a g l
se m
g l a
a
m a s
.co
mThe la e agl
l a se f jalebis
g
p t s o

a The plates of jalebis Gopal bought.

co m
m .
A NSW E R m as e
.co a g l
as emre are 24 jalebis.
The
g l
a  Every plate has jalebis 4 .

 There are 6 plates.
se m
com g l a
 Skip count by 4  4, 8, 12, 16, 20, 24.
m . a
:
ase
Why The plates are equagall So we say one number for each plate instead of
. 24

numbers.

co m
m .
m as e
.co a g l
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g l a
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m .
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Q3 Counting in groups we see there are six groups of four jalebis each or
, , ,4  4  4  4 

4  4  ....... or ........  4  ........ jalebis.

The plates of jalebis
The plates of jalebis Gopal bought.

Six groups of four make 24.

4 + 4 + 4 + 4 + 4 + 4 = 24

6 × 4 = 24 jalebis

Why The tells how many groups The tells how many are in each group
: 6 . 4 .

Q4 Are there enough jalebis for everyone in Dhara s family to have four each Share your
ʼ ?

thoughts in the class.

The plates of jalebis
The plates of jalebis Gopal bought.

No there are not enough jalebis
, .

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9 members, 4 jalebis each

9 × 4 = 36 jalebis needed

They have only 24.

Why : 24 jalebis give 4 each to only 6 people. Three people would get none.

Q5 How many jalebis should Dhara buy so that everyone can get four each ?

The plates of jalebis
The plates of jalebis Gopal bought.

She should buy 12 more jalebis.

Needed: 9 × 4 = 36
Already bought: 24

36 – 24 = 12 more

Why Three people still have none Each needs
: . 4, and 4  4  4  12.

Plants in the arden — Page 93
G

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Dhara and Gopal count the plants
MATH TALK

Q1 Dhara The number of plants is
: i es 6  , 8  6 
6  6  6  6  6  6  6  6  ..... , 8 t m

The flower bed
The flower bed Dhara and Gopal saw on the way home.

Dhara counts the plants column by column .

6 + 6 + 6 + 6 + 6 + 6 + 6 + 6 = 48

8 times 6 = 48

8 × 6 = 48

− Exp anat on
l i

Why The bed has columns Every column has plants so she adds eight
: 8 . 6 , 6

times.

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Q2 Gopal No it is
: , 8  8  8  8  8  8  6  8. Who do you think is correct?

The flower bed
The flower bed Dhara and Gopal saw on the way home.

Both are correct. Both get 48 plants.

Dhara: 8 × 6 = 48

Gopal: 8 + 8 + 8 + 8 + 8 + 8 = 48, so 6 × 8 = 48

About the book: Both dot pictures say Dhara The second one“ ˮ. 6  8 is Gopalʼs.

− Exp anat on
l i

Why Dhara counts columns of Gopal counts rows of It is the same
: 8 6. 6 8.

garden, so the answer is the same.

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co m
m.
Q3 Can you complete this equal grouping and write it as multiplication
e
?

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

om a s
. c
emw t t rst two roups a r a y r n agl
The
a s
sam ots

agl
e d i h he fi g l e d i ged.

co m
m .
m as e
l
Ring the dots in groups of 4  two dots across and two down.
.co a g
a s em  Each ring holds 4 dots.

agl  There are 12 such rings in all.

se m
com g l a
. a
12 × 4 = 48
m
lase
Why No dot is left outaagnd none is counted twice Only the size of the group has
: .

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

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Q4 Can you find more equal groups of different sizes Draw them and write as
?

multiplication.

The dots printed in the book, drawn again.

Yes. The same 48 dots can be grouped in many ways.

GROUP SIZE NUMBER OF GROUPS MULTIPLICATION

2 24 24 × 2 = 48

3 16 16 × 3 = 48

6 8 8 × 6 = 48

8 6 6 × 8 = 48

Why The dots are the same Bigger groups mean fewer groups
: . .

Visit to a Farm — Page 94

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Counting the chickens in twos

Q1 Let us count chickens in the farm! 4 times 2 is ....... or, 4  2  ....... ; 5 times 2 is .......
or, 5  2  ....... ; 6 times 2 is ....... or, 6  2  ....... ; ....... times 2 is ....... or, .......  2 

....... ; ....... times ....... is ....... or, 8  2  ....... ; ....... times ....... is ....... or, 9  2  ....... ;

....... times ....... is ....... or, 10  2  .......

The chicks in the farm, in pairs

The chicks in the farm, standing in pairs.

The chicks stand in pairs. Each new row has one pair more.

Page 31 of 87

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IN WORDS IS MULTIPLICATION

4 times 2 8 4×2=8

5 times 2 10 5 × 2 = 10

6 times 2 12 6 × 2 = 12

7 times 2 14 7 × 2 = 14

8 times 2 16 8 × 2 = 16

9 times 2 18 9 × 2 = 18

10 times 2 20 10 × 2 = 20

Why Every row adds one more pair and a pair is
: , 2. So each answer is 2 more than
the one above it.

Skip Jumping Game — Page 95
Dhara skip jumps by 3
TRY THIS

Q1 Starting from 0, Dhara jumps to 3. From 3 she goes to 6. From 6 she goes to 9. Now,
continue to see how Dhara jumps after 9.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

The number track with Dharaʼs first three jumps drawn on it.

After she lands on 12 then
9 , 15, then 18.

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9 + 3 = 12

12 + 3 = 15

15 + 3 = 18

Why Every jump is of the same size
: , 3 steps. So we keep adding 3.

Q2 DHARA IS SKIP JUMPING BY 3. 
Number of jumps Number reached. 4 jumps ......... →
,5 jumps → ......... , 6 jumps → ......... , 7 jumps → →
......... , 8 jumps ......... , 9 jumps →
......... , 10 jumps → .........

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

The number track with Dharaʼs first three jumps drawn on it.

Each jump adds one more 3.

JUMPS ADDITION NUMBER REACHED

4 3+3+3+3=4×3 12

5 5×3 15

6 6×3 18

7 7×3 21

8 8×3 24

9 9×3 27

10 10 × 3 30

Page 33 of 87

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Why The number of jumps tells how many s we have added That is why.ciot ims the
m
:

se
3 .

com a
times-3 table.
. ag l
m
ase
agl
Let us Do — Pages 96–97
co m
. ag
Skip jumping and times tables

se m
l a
Q1 agber she will jump onto
Guess and write the next num .

co m
em.
m l as
m .co a g
l a se 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
a g
m a s
.co agl
The number track with Dharaʼs first three jumps drawn on it.

se m
g l a
a

com
.
She has reached 9. The next number is 12.
e m
m l as
9 +.c3o= 12 a g
m
ase
agl
Why Every jump is steps long So we add to where she is standing se m
com a
:

l
3 . 3 .

. ag
m
ase
agl

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

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m .
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.co

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Q2 Is there a pattern in these numbers : 3, 6, 9, ...?

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

The number track with Dharaʼs first three jumps drawn on it.

Yes, there is a pattern.
Each number is more than the one before it
3 .

3, 6, 9, 12, 15, 18, 21, 24, 27, 30.

These are exactly the numbers of the times-3 table.

Why Every jump adds one more So the numbers grow by each time
: 3. 3 .

Q3 How many steps forward is Dhara jumping each time ?

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

The number track with Dharaʼs first three jumps drawn on it.

Dhara jumps 3 steps each time .

0→3→6→9

3 – 0 = 3, 6 – 3 = 3, 9 – 6 = 3

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Why The gap between two landing numbers is always That gap is the size of
: 3.

one jump.

Q4 Continue skip jumping by by drawing the jumps on the number track
6, .

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

The number track from 0 to 25, with the first two jumps of 6 drawn on it.

Start at 0 and add 6 each time.

0 → 6 → 12 → 18 → 24

Draw two more jumps : 12 to 18 and 18 to 24. The track ends at 25, so stop at 24.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

The jumps of 6 drawn up to 24 (new jumps in green).

− Exp anat on
l i

Why Each jump is the same size
: , 6 steps. Adding 6 again and again gives the
landing numbers.

Page 36 of 87

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Q5 Can this skip jumping be used to form times -6 t able? Write times-6 table in your
notebook.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

The number track from 0 to 25, with the first two jumps of 6 drawn on it.

Yes. The landing numbers make the times-6 table.

166 6  6  36

2  6  12 7  6  42

3  6  18 8  6  48

4  6  24 9  6  54

5  6  30 10  6  60

Why After jumps we have added four times and that is
: 4 6 , 4  6.

Q6 Is there repeated addition happening Make times ? -4 t able using repeated addition in
the picture given below.

4 4 4


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
14 24

The number track with the first two jumps of 4 drawn on it.

Yes, 4 is being added again and again.

Page 37 of 87

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JUMPS REPEATED ADDITION NUMBER REACHED

1 4 4

2 4+4 8

3 4+4+4 12

4 4+4+4+4 16

5 5×4 20

6 6×4 24

7 7×4 28

8 8×4 32

9 9×4 36

10 10 × 4 40

Why Multiplication is a short way of writing the same addition
: . 3  4 just means 4 
4  4.

Q7 Gopal is doing skip jumping of eps. After 27 he will jump on .............,
............. st

..............

0 5 9 10 15 18 20 25 27 30 35 40 45 50

The number track with Gopalʼs jumps drawn on it.

Gopal is skip jumping of 9 steps .

0 → 9 → 18 → 27

9 – 0 = 9, 18 – 9 = 9, 27 – 18 = 9

After 27 he will jump on 36, then 45.

Page 38 of 87

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co m
m.
− Exp anat on
l i

m as e
. c o a g l
WheymAll his jumps are equal So we keep adding and 36  9 
as
: . 9  27  9  36

a g l 45.

co m
m . ag
l a se
Q8 What times table can you ca g ct from Gopal s jumps Make it in your notebook
onstru ʼ ? .

co m
e m.
m l as
.co a g
0 5 9 10 15 18 20 25 27 30 35 40 45 50

se m
g l a
a
The number track with Gopalʼs jumps drawn on it.

m a s
m .co agl
se
g l a
We get the times-9 table. a
199 6  9  54
co m
m .
m as e
.co l
2  9  18 7  9  63
a g
a s e3m9 27
gl
  8  9  72
a
m
4  9  36 9  9  81

a se
com g l
m. a
5  9  45 10  9  90

ase
: agel more Three jumps mean
Why Each jump adds on 9. 9  9  9, which is 3  9.

com
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 .
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.co

a g l Page 39 of 87

l h h e k h dhan A aS m · NCERT So ut ons
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Q9 Dhara also skip jumps Gopal notes down the jumps but he misses the first few
.

numbers.

32, 40, 48, 56

The numbers Gopal wrote down. The first three boxes are empty.

By what numbers was Dhara skip jumping Construct the times table of this number in
?

your notebook.

Dhara was skip jumping by 8 .

40 – 32 = 8

48 – 40 = 8

56 – 48 = 8

The missed numbers are 8, 16 and 24. The table is:

188 6  8  48

2  8  16 7  8  56

3  8  24 8  8  64

4  8  32 9  8  72

5  8  40 10  8  80

− Exp anat on
l i

Why The gap between two landing numbers is the size of one jump Here the
: .

gap is 8 every time.

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Let us Play — Page 97
Reaching the flower on 12

TRY THIS

Q1 Atya places a flower on Skip jump with equal steps to reach the flower. No direct
12.

jumping to the flower is allowed. The one who reaches the flower in the smallest
number of jumps wins. What skip jumping number will you choose? ..........

12

Atyaʼs flower stands on 12
The playing track with Atyaʼs flower placed on it.

I will choose 6 I reach
. 12 in only 2 jumps.

JUMP SIZE LANDING NUMBERS JUMPS NEEDED

2 2, 4, 6, 8, 10, 12 6

3 3, 6, 9, 12 4

4 4, 8, 12 3

6 6, 12 2

Why A jump of
: 12 would land on the flower straight away, and that is not allowed.
So 6 is the best.

Page 41 of 87

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Q2 Are there numbers that can be reached only through skip jumping by Find 1? 3 su ch
numbers.

0 5 10 15 20 25 30 35 40 45 50

The number track from 0 to 50.

Yes. Three such numbers are 7, 11 and 13.
7  jumps of 2, 3, 4, 5 or 6 all miss it.
11  no equal jump lands on it except 1.

13  the same thing happens.

− Exp anat on
l i

Why A jump of would land on in one go but direct jumping is not allowed
: 7 7 , .

So only jumps of 1 are left.

D you now?
id k 17, 19 and 23 also work the same way.

Q3 Play this game with your friends by putting the flower on different numbers on the
track. See who is able to reach in the minimum number of jumps.

Sample answer: Choose the biggest jump that lands on the flower but is smaller than the
flowerʼs number.

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FLOWER ON BEST JUMP JUMPS NEEDED

10 5 2

15 5 3

18 9 2

20 10 2

Tip: An even number can always be reached in 2 jumps. Jump half of it each time.

Fun Way of Writing Tables — Page 98

Page 43 of 87

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Mithuʼs sticks
co m
e m.
as
TRY THIS
m l
.co a g
em
s ithu figures out another way of writing multiplication tables by drawing sticks! Do
Q1 laM
ag you see repeated addition in this ?

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
a g

co m
m .
m as e
.co a g l
se m
g l a
a
se m
com g l a
m . a
M e
asithuʼs sticks
agl
m
The sticks Mithu arranged, with a red dot at every crossing.
. co
em
m l as
.co
mA NSW E R a g
s e
agla Yes. Each new standing stick adds 5 more red dots.
.c
s e m
a
 Mithu has 5 lying sticks and 4 standing sticks.
yim
 Each standing stick crosses all 5.clo ng sticks and makes 5 dots.
m agl
l a se
a g

co m
m .
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.co

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5 + 5 + 5 + 5 = 20 dots

4 × 5 = 20

Q2 Complete the times -5 t able using sticks. 1  5  5, 2  5  10, 3  5  15, 4  5  , 5  5
= ,6  5  ,7  5  ,8  5  ,9  5 

1 times 5

2 times 5

3 times 5

The first three rows of the times-5 table made with sticks.

Add one more standing stick each time It crosses the lying sticks and makes more red
. 5 5

dots.

4 × 5 = 20

5 × 5 = 25

6 × 5 = 30

7 × 5 = 35

8 × 5 = 40

9 × 5 = 45

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− Exp anat on
l i

Why Each answer is more than the one above One extra stick makes extra
: 5 . 5

dots.

Q3 Make times-6 to times-10 tables using the sticks method shown above.

M
ithuʼs sticks

The sticks Mithu arranged, with a red dot at every crossing.

For the times table keep lying sticks for times
-6 6 , -7 keep 7, and so on. Add standing sticks one
by one and count the red dots.

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STANDING STICKS ×6 ×7 ×8 ×9 × 10

1 6 7 8 9 10

2 12 14 16 18 20

3 18 21 24 27 30

4 24 28 32 36 40

5 30 35 40 45 50

6 36 42 48 54 60

7 42 49 56 63 70

8 48 56 64 72 80

9 54 63 72 81 90

10 60 70 80 90 100

Why The number of lying sticks tells the table Each standing stick adds that many
: .

dots.

Seeing Patterns in Multiplication Tables — Page 99

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The times-5 table

MATH TALK

Q1 Look at the times-5 table. What patterns do you see?

ROW ANSWER

15 5

25 10

35 15

45 20

55 25

65 30

75 35

85 40

95 45

10  5 50

The times-5 table has a clear pattern.
The answers end in 5, 0, 5, 0, 5, 0 …
Odd rows 1, 3, 5, 7, 9 end in 5.
Even rows end in
2, 4, 6, 8, 10 0.

Each answer is more than the one above it
5 .

− Exp anat on
l i

Why Two fives make ten and a ten always ends in
: , 0. One five more brings
back a 5.

Page 48 of 87

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co m
m.
Q2 Guess what will be the last digits of and 12  5.
e
11  5

com g l as
m . ANSWER a
ase
ROW

agl 15 5

co m
10

ag
25

.
m15
as e
l
35

45 a g 20

co m
m.
55 25

m as e
.co
65 30
a g l
se m
g l a
a
75 35

s
85 40

m a
m .co45 agl
se
95

g l a
10  5
a 50

co m
m .
m t of ase
codigi
The la.st a g l
em
11  5 is 5. The last digit of 12  5 is 0.
a s
a gl
m
11 × 5 = 55

a se
. com g l
12 × 5 = 60
m a
ase
− Exp anat on
l i agl

Why ve.cn number, so its
is an odd number, so its answer ends in 5. 12 is an em
om
e
las
: 11

. c oamnswer ends in 0. ag
s e m
a
agl c
m .
m a s e
e m . co agl
g l as
a

co m
m .
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.co

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Q3 Give examples of numbers that when taken
3 i es gives an answer ending with (a)
5 t m

0 ....... ....... ....... ( b) 5 ....... ....... .......

ROW ANSWER

15 5

25 10

35 15

45 20

55 25

65 30

75 35

85 40

95 45

10  5 50

Even numbers give an answer ending in Odd numbers give an answer ending in
0. 5.

ANSWER ENDS WITH NUMBERS CHECK

(a) 0 2, 4, 6 2 × 5 = 10, 4 × 5 = 20, 6 × 5 = 30

(b) 5 1, 3, 7 1 × 5 = 5, 3 × 5 = 15, 7 × 5 = 35

− Exp anat on
l i

Why An even number of fives can be paired into tens so nothing is left over
: , .

An odd number leaves one 5 behind.

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Q4 Without finding the answer, can you tell the last digits of 18  5, 23  5, 32  5, 50  5.

ROW ANSWER

15 5

25 10

35 15

45 20

55 25

65 30

75 35

85 40

95 45

10  5 50

Yes. Just look at whether the number is even or odd.

PRODUCT EVEN OR ODD LAST DIGIT

18  5 18 is even 0

23  5 23 is odd 5

32  5 32 is even 0

50  5 50 is even 0

Why In the times table the last digit is always or Even numbers give and
: -5 0 5. 0

odd numbers give 5.

-
In text Questions — Page 100

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How the tables of 2, 3 and 5 are joined
MATH TALK

Q1 Look at the times tables of 2, 3, 5. They have a relation between them. Can you see it?

122 133 155

224 236 2  5  10

326 339 3  5  15

428 4  3  12 4  5  20

The first four rows of the times-2, times-3 and times-5 tables
The three table cards printed in the book.

Yes. The times-2 answer plus the times-3 answer gives the times-5 answer.

2+3=5

4 + 6 = 10

6 + 9 = 15

8 + 12 = 20

Why Taking a number times and then times means taking it times in all
: 2 3 5 .

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Q2 Is there a relation between the two circled numbers and the boxed number Does this ?

happen for the next rows also?

122 133 155

224 236 2  5  10

326 339 3  5  15

428 4  3  12 4  5  20

The first four rows of the times-2, times-3 and times-5 tables
The three table cards printed in the book.

Yes. The two circled numbers add up to the boxed number.

Circled: 2 and 3. Boxed: 5.

2+3=5

It happens in the next rows too : 4  6  10, 6  9  15, 8  12  20.

− Exp anat on
l i

Why In every row
: , 1  2 and 1  3 join to make 1  5, and so on down the table.

Page 53 of 87

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co m
m.
Q3 Can you find other examples of two tables adding up to a third table
e
?

m l as
m.co a g
l a se
a g 122 133 155

com
224 236 2  5  10

e m . ag
s
326 339 3  5  15
a 4
428 agl  3  12 4  5  20

oms
The first four rows of the times-2, times-3 and times-5 table
. c
s e m
m a
.co agl
The three table cards printed in the book.

se m
g l a
a
m a s
.co agl
Yes, there are many such pairs.

a s em
TWO TABLES
a RlD TABLE
THIg EXAMPLE

m
2 and 2 4 6 + 6 = 12, and 3 × 4 = 12

9 + 12 = 21, and 3 × 7m
.co
m a se
3 and 4 7 = 21

. co l
agand 2 × 9 = 18
m
ase
4 and 5 9 8 + 10 = 18,

agl 6 and 4 10 18 + 12 = 30, and 3 × 10 = 30

se m
com g l a
Why Take a number times aenmd.then times You have taken it times in all a
s
: 3 4 . 7 .

a
agl
Let us Do — Page 100 co m
m .
m as e
l
Word problems on multiplication and division
.co a g
a s em
agl Q1 Draw pictures for each of the following problems in your notebook Use counting . ,

number line jumps or any other method to solve the problems. (a) There are 5 jars with
.c
cookies in each jar. How many cookies are there?
s e m
m a
co agl
4

m .
as e
a g l
There are 20 cookies.

co m
m .
m as e
.co

a g l Page 54 of 87

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4 + 4 + 4 + 4 + 4 = 20

5 × 4 = 20

Why There are equal groups of
: 5 4. Skip count by 4  4, 8, 12, 16, 20.

Q2 (b) An idli vessel contains 6 idli plates. In each plate we can make 4 idlis. How many
idlis can be cooked in one go?

24 idlis can be cooked in one go.

4 + 4 + 4 + 4 + 4 + 4 = 24

6 × 4 = 24

Why Each plate holds the same idlis Six plates mean six fours
: 4 . .

Q3 c
( ) 30cookies are to be distributed among 5 children equally. How many cookies will
each child get?

Each child gets 6 cookies .

30 – 5 = 25 25 – 5 = 20 20 – 5 = 15

15 – 5 = 10 10 – 5 = 5 5 – 5 = 0

We took 5 away 6 times, so 30 ÷ 5 = 6.

− Exp anat on
l i

Why Each round gives cookie to all children There were rounds
: 1 5 . 6 .

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Q4 d) Roro starts from 0 and takes 6 jumps to reach 18. All his jumps are of the same size.
(

What is the size of Roroʼs jump?

Each jump is of 3 steps .

0 → 3 → 6 → 9 → 12 → 15 → 18

18 ÷ 6 = 3

Why Six equal jumps make
: 18. Since 6  3  18, one jump must be 3.

Q5 e Toto does not take jumps of the same size and still reaches 18 in 6 jumps. How did
( )

Toto jump?

Totoʼs jumps were of different sizes, but they still add up to 18.

One way: 2 + 2 + 3 + 3 + 4 + 4 = 18

Another way: 1 + 2 + 3 + 4 + 5 + 3 = 18

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l i

Why Any numbers that add up to
: 6 18 will do. Only the total must be 18.

Q6 f Suma saves ₹ 8 every day. After how many days will she have ₹ 56?
( )

After 7 days she will have ₹ 56.

8, 16, 24, 32, 40, 48, 56

7 × 8 = 56, so 56 ÷ 8 = 7

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l h h e k h dhan A aS m · NCERT So ut ons
gl e l i

Why Each day adds one more
: ₹ 8. We count how many 8s make 56.

Q7 (g) Mary has 63 sea-shells. She gives 7 sea-shells to each of her 5 friends. How many
does she have left?

She has 28 sea-shells left.

Given away: 5 × 7 = 35

Left: 63 – 35 = 28

Why First find how many she gave in all Then take that away from
: . 63.

Q8 Solve the following problems. Try constructing a word problem. (a) 4  9

4 × 9 = 36

My word problem: There are rows of chairs Each row has chairs How many chairs are
4 . 9 .

there?

Why : 9  9  9  9  36. Four equal groups of 9 make 36.

Q9 ( b) 32  8

32 – 8 = 24 24 – 8 = 16

16 – 8 = 8 8 – 8 = 0

32 ÷ 8 = 4

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l h h e k h dhan A aS m · NCERT So ut ons
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My word problem: 32 mangoes are shared equally among 8 children. How many does each
child get?

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Why We took away four times before nothing was left
: 8 .

Q10 c
( ) 6  7

7 + 7 + 7 + 7 + 7 + 7 = 42
6 × 7 = 42

My word problem: A week has days How many days are there in weeks
7 . 6 ?

Why Skip count by
: 7  7, 14, 21, 28, 35, 42.

Q11 d) 45  5
(

45, 40, 35, 30, 25, 20, 15, 10, 5, 0

We took 5 away 9 times.

45 ÷ 5 = 9

My word problem: 45 laddoos are put equally in 5 boxes. How many laddoos go in each box?

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l i

Why : 9  5  45, so 45 shared by 5 is 9 each.

Page 58 of 87

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l h h e k h dhan
a g l A aS m · NCERT So ut ons
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- — Page 101 co m
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