NCERT Solutions Class 5 Maths Chapter 6 the Dairy Farm

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 5 · M AT H S

NCERT Solutions

Chapter 6: The Dairy Farm

NCERT Textbook — Math Mela

BOOK PAGES SECTIONS QUESTIONS MEDIUM

70 – 91 18 95 English

Solutions, notes, sample papers & more at 98 pages

Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

CLASS 5 · MATHS · MATH MELA

NCERT Solutions — Chapter 6: The Dairy Farm
Chapter 6 of Math Mela is the longest chapter in the book, and it is all about multiplication. Daljeet Kaur's
butter packets, a family's kitchen waste, a dairy cooperative with 268 villagers and a king who rewards three
ministers with gold coins teach one big idea: there is never only one way to multiply. You can change the
order, double and halve, jump to the nearest multiple, or break a number into its place values — and
every road ends at the same product.

TEXTBOOK BOOK PAGES

Math Mela (Class 5) 70 – 91

SECTIONS QUESTIONS

18 95

MEDIUM

English

Let Us Think — Pages 70–71
Shape puzzles, butter milk pouches, Which number am I?, Make your own numbers

LET US THINK

Q1 The given shapes stand for numbers between 1 and 24. The same shape denotes the
same number across all problems. Find the numbers hiding in all the shapes.

a) × = d) × =

b) × = e) × =

c) × = f) × = ×

The six shape problems printed on page 70. The same shape always stands for the same
number.

One set of numbers that works: green pentagon = 1, orange diamond = 2, pink circle = 3, blue
rectangle = 4, yellow parallelogram = 6, purple triangle = 12, white shape = 16, blue half-circle =
18.

Page 1 of 98

Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

Here is how to unlock the puzzle, step by step. Start with the clue that gives away a number all
by itself.

1. Look at (d): diamond × pentagon = diamond. A number stays the same only when you
multiply it by 1. So the green pentagon = 1.
2. Look at (b): rectangle × rectangle = white shape. The rectangle times itself must stay
below 24. So the rectangle can only be 1, 2, 3 or 4. Take the biggest, rectangle = 4. Then
white shape = 4 × 4 = 16.
3. Look at (a): rectangle × circle = triangle. Try circle = 3. Then triangle = 4 × 3 = 12.
4. Look at (c): parallelogram × diamond = triangle. We need two numbers whose product is
12. Take parallelogram = 6 and diamond = 2. Check (d) again: 2 × 1 = 2 ✓.
5. Look at (e): parallelogram × circle = half-circle. 6 × 3 = 18. That is under 24, so it is allowed.
6. Check (f): half-circle × rectangle = 18 × 4 = 72, and parallelogram × triangle = 6 × 12 = 72.
Both sides give 72 ✓.

SHAPE NUMBER CHECK

Green pentagon 1 (d) 2 × 1 = 2

Orange diamond 2 (c) 6 × 2 = 12

Pink circle 3 (a) 4 × 3 = 12

Blue rectangle 4 (b) 4 × 4 = 16

Yellow parallelogram 6 (e) 6 × 3 = 18

Purple triangle 12 (f) 6 × 12 = 72

White shape 16 (b) 4 × 4 = 16

Blue half-circle 18 (f) 18 × 4 = 72

Why it happens: Clue (d) is the key that opens the door. Multiplying by 1 never
changes a number, so the pentagon had to be 1. After that, every other clue only had
to agree with the ones before it.

Did you know? This puzzle has more than one answer. If you take the rectangle as 3
instead of 4, other sets of numbers also fit. Try it and see what you get — just make
sure every shape stays between 1 and 24.

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Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

Q2 Place the digits 2, 5, and 3 appropriately to get a product close to 100. Share your
reasoning in class.

Put 53 in the two-digit box and 2 in the one-digit box: 53 × 2 = 106. That is the closest you can
get to 100.
There are only six ways to arrange the three digits. Try them all and compare.

ARRANGEMENT PRODUCT HOW FAR FROM 100?

53 × 2 106 6

23 × 5 115 15

52 × 3 156 56

32 × 5 160 60

25 × 3 75 25

35 × 2 70 30

53 × 2

Step 1. 50 × 2 = 100

Step 2. 3 × 2 = 6

Step 3. 100 + 6 = 106

Sample reasoning to share in class: I want an answer near 100. If the one-digit
number is big, like 5, the answer shoots past 100. So I kept the small digit 2 as the
multiplier. Then I made the two-digit number as close to 50 as I could, because 50 ×
2 is exactly 100. The biggest number I could build from 5 and 3 is 53, and 53 × 2 =
106. Only 6 away.

Page 3 of 98

as e
Class 5 Maths Chapter 6 The Dairy Farm
a g l AglaSem · NCERT Solutions

co m
m.
A dairy has packed butter milk pouches in the following manner. Find the number
e
Q3

com l as
of pouches kept in each arrangement. One is done for you. (First arrangement: 30 ×
g
. a
em
2 = 60.)

a s
a gl

m
All four arrangements hold 60 pouches. Only the way they are grouped changes.
. c o ag
s
1. First arrangement (already done): pouches
e m sit in pairs. There are 6 pairs in each row and 5
rows, so 6 × 5 = 30 pairs. That is 30l×a2 = 60.
g
2. Second arrangement: pouchesasit in threes. There are 4 threes in each row and 5 rows, so 4

m
× 5 = 20 groups. That is 20 × 3 = 60.
co
m.
3. Third arrangement: tall blocks of 2 across and 5 down, so 10 pouches in each block. There

o m
are 6 blocks. That is 6 × 10 = 60.
l a se
g That is 5 × 12 = 60.
.carrangement: long rows of 12 pouches, and there are 5 arows.
se m
4. Fourth

g l a
a
Same 60 pouches, four ways to group them
m a s
m .co agl
l a se
groups of 2 → 30 ×a 2 g groups of 3 → 20 × 3

co m
blocks of 10 → 6 × 10
m .
e
.co
m las5 × 12
rows of 12 →
ag
a sem
agl The pouches never change. Only the size of the group changes, so the two numbers swap around.

se m
com g l a
m . a
ase
30 × 2 = 60

20 × 3 = 60 agl
6 × 10 = 60
co m
m .
5 × 12 = 60
m as e
.co a g l
se m
g l a
a Why it happens: When you make the groups bigger, you need fewer of them. Twice
c
the group size means half the number of groups, so the total stays exactly the same.
m .
m a s e
e m . co agl
g l as
a

com
m .
m ase
.co


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Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

Q4 What other groups can you make? (with the butter milk pouches)

You can make twelve different groupings of 60 pouches. Every pair of numbers that
multiplies to 60 gives one.

NUMBER OF GROUPS GROUP SIZE TOTAL

1 60 60

2 30 60

3 20 60

4 15 60

5 12 60

6 10 60

10 6 60

12 5 60

15 4 60

20 3 60

30 2 60

60 1 60

How to find them all without missing any:

1. Start at 1. Ask: does 60 share out evenly into 1 group? Yes — one group of 60.
2. Try 2, then 3, then 4 and so on. Each time, ask whether 60 divides evenly.
3. Write the partner next to it. 60 ÷ 3 = 20, so 3 and 20 are partners.
4. Stop when the pairs start repeating. After 6 × 10 you meet 10 × 6, which you already have.

Check it yourself: 7 does not work, because 60 pouches cannot be shared into
groups of 7 without some left over. Nor do 8, 9, 11, 13 or 14. That is why they are
missing from the table.

Page 5 of 98

Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

Q5 Which number am I? I am a two-digit number. Find me with the help of the
following clues. (a) I am greater than 8. (b) I am not a multiple of 4. (c) I am a
multiple of 9. (d) I am an odd number. (e) I am not a multiple of 11. (f) I am less than
50. (g) My ones digit is even. (h) My tens digit is odd.

The number is 18.
Take the clues one at a time and cross out numbers as you go.

1. Clue (c) — I am a multiple of 9. The two-digit multiples of 9 are 18, 27, 36, 45, 54, 63, 72, 81,
90, 99.
2. Clue (f) — I am less than 50. Only 18, 27, 36, 45 are left.
3. Clue (g) — my ones digit is even. 27 ends in 7 and 45 ends in 5, both odd. Cross them out.
Left: 18, 36.
4. Clue (b) — I am not a multiple of 4. 36 = 4 × 9, so cross out 36. Left: 18.
5. Clue (h) — my tens digit is odd. 18 has 1 in the tens place, and 1 is odd ✓.

Multiples of 9 below 50: 18, 27, 36, 45

Ones digit even → 18, 36

Not a multiple of 4 → 18

One clue does not fit: Clue (d) says the number is odd, but 18 is even. No number
can have an even ones digit (clue g) and still be odd (clue d) — those two clues fight
each other. Every other clue points firmly to 18, so 18 is the answer. If your teacher
asks, say so honestly: clue (d) looks like a printing slip.

Q6 Did you use all the clues to find the number? Which clues did not help you in finding
the number?

No — two clues were of no help at all: (a) and (e).

1. Clue (a) — I am greater than 8. The puzzle already told us the number has two digits. Every
two-digit number is 10 or more, so it is already bigger than 8. This clue removes nothing.
2. Clue (e) — I am not a multiple of 11. The only two-digit multiples of 11 are 11, 22, 33, 44,
55, 66, 77, 88 and 99. None of them is a multiple of 9 except 99, and 99 was already gone
because it is bigger than 50. So this clue removes nothing either.

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Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

The clues that did the work were (c) multiple of 9, (f) less than 50, (g) ones digit even, and (b) not
a multiple of 4.

Why it happens: A clue only helps if it throws some numbers away. If everything left
over already obeys the clue, the clue is just extra words. Good puzzle-solvers always
check which clues are actually doing the work.

Q7 Make your own numbers. Choose any two numbers and one operation from the
grid. Try to make all the numbers between 0 and 20. For example, 2 can be formed
as 4 – 2. Could you make all the numbers?

−
100 25 5
×
10 2 36
+

12 4 3 ÷

The number grid and the four operation cards printed on page 71.

You can make every number from 1 to 20 except 19. The number 0 cannot be made either.
So 19 numbers out of 21 are possible.
Here is one way to make each one. There are often several.

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Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

NUMBER ONE WAY NUMBER ONE WAY

0 not possible 11 36 − 25

1 3−2 12 4×3

2 4−2 13 3 + 10

3 5−2 14 4 + 10

4 12 ÷ 3 15 5 + 10

5 3+2 16 4 + 12

6 2×3 17 5 + 12

7 5+2 18 36 ÷ 2

8 4×2 19 not possible

9 36 ÷ 4 20 5×4

10 12 − 2

Why 0 is impossible: To get 0 by subtracting you need two equal numbers, and
every number in the grid is different. To get 0 by multiplying you need a 0 in the grid,
and there is none. Adding and dividing two positive numbers can never give 0 either.

Try this: Make a list of every answer you can reach with just the small numbers 2, 3,
4, 5, 10 and 12. You will find most of 0 to 20 hiding there.

Q8 Which numbers could you not make? Is it possible to make these numbers using
three numbers? You can use two operations, if needed.

The numbers you could not make with two grid numbers are 0 and 19. With three
numbers and two operations, both become easy.

Page 8 of 98

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Class 5 Maths Chapter 6 The Dairy Farm
a g l AglaSem · NCERT Solutions

co m
e m.
0=5−3−2
m l as
.co
Step 1. 5 − 3 = 2
m a g
l a se
g
Step 2. 2 − 2 = 0
a

co m
. ag
19 = 12 + 10 − 3
em
Step 1. 12 + 10 = 22
g l as
Step 2. 22 − 3 = 19 a

co m
em.
as
Other ways that also work:

co4m− 3, because 12 ÷ 4 = 3 and 3 − 3 = 0.
0 = 12.÷
a g l
a s e=m25 − 4 − 2, because 25 − 4 = 21 and 21 − 2 = 19.
gl
19
a 19 = 36 ÷ 2 + ... no, that gives 18 and there is no 1 in the grid. So this one does not work —

s
always check.
m a
em
.co agl
a s
Why the third number helps: With two numbers you only get one jump. With three
aglso you can overshoot and then come back. That is
numbers you get two jumps,
exactly what 12 + 10 − 3 does: it jumps to 22 and steps back to 19.

co m
m .
o m l a se
m .cWhich numbers between 0–20 can you get in more thanagone way?
e
las
Q9

ag
se m
com g l a
. a
Most of them. Only 11, 16, 17 and 18 have just one way each. Every other number from 1 to 20
m
ase
can be made in two or more ways.

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

com
m .
m ase
.co


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Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

NUMBER WAYS TO MAKE IT

1 3 − 2, 5 − 4, 4 − 3

2 4 − 2, 5 − 3, 12 − 10, 10 ÷ 5, 4 ÷ 2

3 5 − 2, 36 ÷ 12

4 12 ÷ 3, 100 ÷ 25

5 3 + 2, 25 ÷ 5, 10 − 5, 10 ÷ 2

6 4 + 2, 2 × 3, 10 − 4, 12 ÷ 2

7 5 + 2, 3 + 4, 12 − 5, 10 − 3

8 5 + 3, 10 − 2, 4 × 2, 12 − 4

9 12 − 3, 36 ÷ 4

10 12 − 2, 100 ÷ 10, 5 × 2

11 36 − 25 only

12 2 + 10, 4 × 3, 36 ÷ 3

13 3 + 10, 25 − 12

14 2 + 12, 4 + 10

15 5 + 10, 3 + 12, 25 − 10, 5 × 3

16 4 + 12 only

17 5 + 12 only

18 36 ÷ 2 only

20 5 × 4, 25 − 5, 100 ÷ 5, 2 × 10

Check it yourself: The small numbers 2, 3, 4, 5, 10 and 12 pair up in many ways, so
the small answers have many routes. The big numbers 25, 36 and 100 only fit a few
places, so 11, 16, 17 and 18 have just one route each.

Order of Numbers in Multiplication — Page 72

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Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

Daljeet Kaur arranges the butter packets

MATH TALK

Q1 Daljeet Kaur runs a milk processing unit. She has arranged the butter packets in
the following ways. Find the number of butter packets in each case. What pattern
do you notice (or observe)? Discuss in class. (a) 2 × 3 and 3 × 2 (b) 5 × 8 and 8 × 5, then
6 × 13 and 13 × 6, (e) 10 × 5 and 5 × 10 (f) 8 × 20 and 20 × 8 (g) 12 × 9 and 9 × 12

Both arrangements in every pair hold the same number of packets. Here are all the
answers.

CASE FIRST WAY SECOND WAY PACKETS

(a) 2×3 3×2 6

(b) 5×8 8×5 40

(c) 6 × 13 13 × 6 78

(e) 10 × 5 5 × 10 50

(f) 8 × 20 20 × 8 160

(g) 12 × 9 9 × 12 108

The pattern: the number of groups and the group size are swapped, but the total never
changes.

3 rows of 5 = 15 5 rows of 3 = 15

3 × 5 = 15

5 × 3 = 15
Turn the tray a quarter turn. Not one packet moved.

The same tray of packets, seen two ways. Turning it does not add or remove anything.

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Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

Why it happens: Think of the packets laid out in a rectangle. Counting along the
rows gives one multiplication. Counting down the columns gives the other. You are
counting the very same packets, just in a different direction, so the total must be
equal.

Sample answer for the class discussion: We noticed that 6 × 13 and 13 × 6 both
give 78. So the order of the two numbers does not matter when we multiply. This is
handy — 8 × 20 is easier to work out than 20 × 8 in your head, and you are allowed
to choose whichever is easier.

Q2 What is 9 × 0? 0 × 9? Is this true for the product of any two numbers? Discuss in
class.

9 × 0 = 0 and 0 × 9 = 0. Yes, the order rule is true for every pair of numbers.

1. Read 9 × 0 as nine groups of nothing. Nine empty trays hold no packets at all. So 9 × 0 = 0.
2. Read 0 × 9 as no groups of nine. There are no trays, so again there are no packets. So 0 × 9
= 0.
3. Compare. Both give 0, so 9 × 0 = 0 × 9.

9×0=0

0×9=0

So 9 × 0 = 0 × 9 ✓

Why it is true for any two numbers: Any multiplication can be drawn as a rectangle
of dots. Swapping the two numbers only turns the rectangle on its side. The dots are
the same dots, so the answer is the same answer. Mathematicians call this the order
property of multiplication.

Sample answer for the class discussion: We tested it with big numbers too. 25 × 4
= 100 and 4 × 25 = 100. It worked every time. We also noticed it does not work for
subtraction: 9 − 4 = 5 but 4 − 9 is not 5. So the order rule belongs to multiplication
and addition, not to subtraction and division.

Page 12 of 98

Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

Patterns in Multiplication by 10s and 100s — Pages 72–74
Filling the place value tables

LET US DO

Q1 Let us revise multiplication by 10s and 100s. a) 4 × 10 = ___ b) 20 × 10 = ___ c) 10 × 40 =
___ d) 10 × 10 = 100 e) 20 × 50 = ___ f) 80 × 10 = ___ g) 3 × 100 = 100 × 3 = 300 h) 8 × 100 =
___ = ___ i) 10 × 100 = ___ = ___

Here are all the answers.

QUESTION ANSWER THINK OF IT AS

a) 4 × 10 40 4 tens

b) 20 × 10 200 20 tens

c) 10 × 40 400 40 tens

d) 10 × 10 100 10 tens (given)

e) 20 × 50 1,000 2 × 5 × 10 × 10 = 10 × 100

f) 80 × 10 800 80 tens

g) 3 × 100 100 × 3 = 300 3 hundreds (given)

h) 8 × 100 100 × 8 = 800 8 hundreds

i) 10 × 100 100 × 10 = 1,000 10 hundreds

To multiply by 10

Step 1. Take the number.

Step 2. Shift every digit one place to the left.

Step 3. Write 0 in the empty ones place.

Example: 4 → 40, 20 → 200, 80 → 800

Page 13 of 98

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Class 5 Maths Chapter 6 The Dairy Farm
a g l AglaSem · NCERT Solutions

co m
m.
Why the zero appears: Multiplying by 10 makes a number ten times bigger. A digit

m l a se
that was worth ones is now worth tens, and a digit that was worth tens is now worth
o g is left empty
.cEvery digit moves one house to the left, and the onesahouse
m
hundreds.
—a sose
a g l we write 0 there to hold the place.

o m
c ag
For 100, shift twice. 8 × 100: the 8 moves from the ones place to the hundreds
.
mFor 1,000, shift three times.
s e
place, and two zeros fill the gap — 800.
a
agl

co m
m.
Q2 Find answers to the following questions. 100 × 90 = __,000; 400 × 10 = ___; 60 × 50 =

m
____; 30 × 20 = 600; 700 × 4 = _,_00; 10 × 45 = ____
as e
.co a g l
se m
g l a

a All six answers, with the easy way to get each one.
m a s
QUESTION BREAK IT UP
m .co ANSWER
agl
l a se
100 × 90
a g
1 × 9 × 100 × 10 = 9 × 1,000 9,000

m
400 × 10 4 × 100 × 10 = 4 × 1,000 4,000

. co
e m
as
60 × 50 6 × 5 × 10 × 10 = 30 × 100 3,000

com
30 ×.20 a g l
m
ase
3 × 2 × 10 × 10 = 6 × 100 600 (given)

agl 700 × 4 7 × 4 × 100 = 28 × 100 2,800

se m
10 × 45 45 tens
com
450
g l a
m . a
ase
60 × 50 agl
o m
Step 1. Set the zeros aside: 6 × 5 = 30
. c
s e m zeros.
Step 2. Count the zeros you set aside: one from 60, one from 50, so two

com3. Put both zeros back: 3,000 a
m .Step agl
l a se
ag
.c
Why this works: 60 is 6 × 10 and 50 is 5 × 10. Because we may multiply in any order,
s e m
m a
. co agl
6 × 10 × 5 × 10 can be rearranged as 6 × 5 × 10 × 10, which is 30 × 100 = 3,000.
e m
g l as
a

co m
m .
m ase
.co


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Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

Look out: the table on the same page asks for 700 × 8, while the bubble asks for 700
× 4. Both are worth doing: 700 × 4 = 2,800 and 700 × 8 = 5,600.

Q3 How should we write 450 in the table below?

PROBLEM TH H T O

10 × 45 =

30 × 20 = 6 0 0

400 × 10 =

700 × 8 =

100 × 90 = 9 0 0 0

The place value table printed on page 73. Two rows are already filled in.

Write 4 in the hundreds column, 5 in the tens column and 0 in the ones column. The
thousands column stays empty.

NUMBER TH H T O

450 4 5 0

1. Read the number from the right. The last digit sits in the ones column. Here it is 0.
2. Move left one column. That is tens. The digit there is 5, so 450 has 5 tens = 50.
3. Move left again. That is hundreds. The digit there is 4, so 450 has 4 hundreds = 400.
4. Check by adding. 400 + 50 + 0 = 450 ✓

Why the ones column needs a 0: If we left it blank we would only see 45, which is a
completely different number. The 0 is a placeholder — it keeps the 4 and the 5 sitting
in their correct houses.

Page 15 of 98

Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

Q4 Fill in the table. Describe the pattern.

PROBLEM TH H T O

10 × 45 =

30 × 20 = 6 0 0

400 × 10 =

700 × 8 =

100 × 90 = 9 0 0 0

The place value table printed on page 73. Two rows are already filled in.

The completed table.

PROBLEM TH H T O ANSWER

10 × 45 4 5 0 450

30 × 20 6 0 0 600 (given)

400 × 10 4 0 0 0 4,000

700 × 8 5 6 0 0 5,600

100 × 90 9 0 0 0 9,000 (given)

700 × 8

Step 1. 7 × 8 = 56

Step 2. 700 has two zeros, so put two zeros back.

Step 3. Answer = 5,600

The pattern: count the zeros in the two numbers you are multiplying, do the small
multiplication first, then write that many zeros after it.

Page 16 of 98

Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

Why it happens: Each zero at the end of a number means it has been multiplied by
10 once more. Multiplying by 10 shifts every digit one place left and leaves a 0
behind. So the zeros simply queue up at the end of the answer.

Q5 Fill in the tables. Describe the pattern. Discuss in class.

PROBLEM TH H T O

60 × 50 =

220 × 20 =

11 × 300 =

PROBLEM TH H T O

80 × 90 =

10 × 63 =

40 × 12 =

The two place value tables printed on page 73.

Both tables, completed.

Page 17 of 98

Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

PROBLEM TH H T O ANSWER

60 × 50 3 0 0 0 3,000

220 × 20 4 4 0 0 4,400

11 × 300 3 3 0 0 3,300

80 × 90 7 2 0 0 7,200

10 × 63 6 3 0 630

40 × 12 4 8 0 480

220 × 20

Step 1. Set the zeros aside: 22 × 2 = 44

Step 2. Zeros set aside: one from 220, one from 20 → two zeros

Step 3. Answer = 4,400

40 × 12

Step 1. 4 × 12 = 48
Step 2. One zero from 40

Step 3. Answer = 480

Why it happens: Every zero on the end is a hidden 10. 220 × 20 is really 22 × 10 × 2 ×
10, and we may multiply in any order. So it is 22 × 2 first, giving 44, and then × 10 ×
10, which pushes 44 two places to the left.

Sample answer for the class discussion: We saw that the answers all end in zeros,
and the number of zeros equals the number of zeros in the question. 60 × 50 has
two zeros in the question and two at the end of 3,000. But watch out — 60 × 50 =
3,000 has three zeros, because 6 × 5 = 30 already brings a zero of its own.

Page 18 of 98

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Class 5 Maths Chapter 6 The Dairy Farm
a g l AglaSem · NCERT Solutions

co m
m.
What will happen if we multiply numbers by 1,000? 2 × 1,000 = ; 5 × 1,000 = ; 10 ×
e
Q6

m l as
.co
1,000 = ; 20 × 1,000 =

a g
se m
g l a
a

Every digit shifts three places to the left, and three zeros appear at the end.

com
e m . ag
as
2 × 1,000 = 2 thousand = 2,000

5 × 1,000 = 5 thousand = 5,000 agl

m
10 × 1,000 = 10 thousand = 10,000
co
e m.
as
20 × 1,000 = 20 thousand = 20,000
m l
m .co a g
l a se
a g
TTh Th H T O
a s
.com agl
m
ase 5
agl
5 0 0 . co0m
sethe
5mjumps three houses to the left. Three zeroslafill
m
o
.c empty houses. ag
se m
g l a
a Multiplying by 1,000 moves each digit three place values to the left.

se m
com g l a
m . a
ase
Why three places: 1,000 is 10 × 10 × 10. Each 10 moves the digits one house left, so

agl
three tens move them three houses left.

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


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Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

Q7 Let us fill in the table and observe the patterns.

PROBLEM TTH TH H T O

2 × 1,000 = 2 0 0 0

5 × 1,000 = 5 0 0 0

10 × 1,000 = 1 0 0 0 0

20 × 1,000 = 2 0 0 0 0

3 × 5,000 =

8 × 3,000 =

5 × 7,000 =

The first table printed on page 74. The first four rows are already filled in.

The completed table.

PROBLEM TTH TH H T O ANSWER

2 × 1,000 2 0 0 0 2,000 (given)

5 × 1,000 5 0 0 0 5,000 (given)

10 × 1,000 1 0 0 0 0 10,000 (given)

20 × 1,000 2 0 0 0 0 20,000 (given)

3 × 5,000 1 5 0 0 0 15,000

8 × 3,000 2 4 0 0 0 24,000

5 × 7,000 3 5 0 0 0 35,000

Page 20 of 98

Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

8 × 3,000

Step 1. 8 × 3 = 24

Step 2. 3,000 has three zeros.

Step 3. Write 24 and add three zeros: 24,000

Read it aloud: 8 × 3 thousand = 24 thousand. Saying the word "thousand" out loud
stops you from losing a zero.

Q8 Fill in the table.

PROBLEM TTH TH H T O

20 × 100 =

40 × 500 =

60 × 300 =

600 × 30 =

80 × 900 =

70 × 600 =

5 × 7,000 =

The second table printed on page 74.

The completed table.

Page 21 of 98

Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

PROBLEM SMALL MULTIPLICATION ZEROS TO ADD ANSWER

20 × 100 2×1=2 3 2,000

40 × 500 4 × 5 = 20 3 20,000

60 × 300 6 × 3 = 18 3 18,000

600 × 30 6 × 3 = 18 3 18,000

80 × 900 8 × 9 = 72 3 72,000

70 × 600 7 × 6 = 42 3 42,000

5 × 7,000 5 × 7 = 35 3 35,000

Notice something lovely: 60 × 300 and 600 × 30 both give 18,000. The zeros just
moved from one number to the other, and the total number of zeros stayed at three.
That is the order rule at work again.

Check it yourself: Count the zeros in the question and the zeros in your answer. In
40 × 500 the question has three zeros, and 4 × 5 = 20 brings one more, so the answer
20,000 has four zeros in all.

Many Ways to Multiply — Page 74
Five children find 18 × 5

MATH TALK

Q1 What is 18 × 5?

18 × 5 = 90.
The five children in the book each found it a different way, and all five are right.

Page 22 of 98

Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

CHILD'S METHOD THE WORKING ANSWER

Halve 18, then double the answer 9 × 5 = 45, and 45 + 45 90

Double, double, then add one more 18 + 18 = 36, 36 + 36 = 72, 72 + 18 90

Split 18 into 10 and 8 10 × 5 = 50, 8 × 5 = 40, 50 + 40 90

Doubling and halving 18 × 5 = 9 × 10 90

Nearest multiple 20 × 5 = 100, take away 2 × 5 = 10 90

The easiest one to do in your head

Step 1. Half of 18 is 9.

Step 2. Double of 5 is 10.

Step 3. 9 × 10 = 90

Try this: Cover the table and find 18 × 5 in a sixth way of your own. For example, 18
× 5 is half of 18 × 10, and 18 × 10 = 180, so the answer is 90.

Q2 Do you think they are all correct? Why do you think so?

Yes, all five are correct. Every one of them ends at 90, and each one is built on a rule we can
trust.

1. Halving and doubling back. 9 is half of 18, so 9 × 5 is half the answer. Adding it twice gives
the whole answer. 45 + 45 = 90 ✓
2. Doubling twice and adding one more group. Two 18s, then four 18s, then one more 18
makes five 18s. 36, 72, then 72 + 18 = 90 ✓
3. Splitting 18 into 10 and 8. Five groups of 18 is the same as five groups of 10 plus five
groups of 8. 50 + 40 = 90 ✓
4. Doubling one and halving the other. Halve 18 to 9 and double 5 to 10. 9 × 10 = 90 ✓
5. Going to the nearest multiple. Five 20s is 100. But we only wanted five 18s, which is five 2s
less. 100 − 10 = 90 ✓

Page 23 of 98

as e
Class 5 Maths Chapter 6 The Dairy Farm
a g l AglaSem · NCERT Solutions

co m
m.
Why they must all agree: The question never changed — it is still five groups of 18

m l a se
packets. The children only chose different ways to count the same packets. Different
o
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roads, same

l a se
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o m
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of them adds anything or takes anything away without putting it back. The method
you pick can depend on the numbers.m .
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a 10.
because it turns a hard 5 into an leasy
For 18 × 5, doubling and halving is fastest,

ag

co m
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coofmbutter packets in half
Doubling and Halving — Pages 75–76
g l as
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Cutting a block

a s
agl
DOUBLING AND HALVING

om a s
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Q1 Butter packets are arranged in the following ways. (a) 3 × 18: Double one, halve the
other, to get 6 × 9 = 54. Discuss m
s e why these are the same.

a gla

co m
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They are the same because no packet is added and none is taken away — the block is only
cut and re-stacked. 3 × 18 = 6 × 9 = 54.
se m
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m .c at the first block. It is 3 packets deep and 18 packets long.
a
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1. Look

a
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se m
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4. Count. 6 × 9 = 54, the same as 3 × 18 = 54.

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3 deep, 18 longag→
ase

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cut here
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a s em stacked: 6 deep, 9 long → 6 × 9
agl
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Both hold 54 packets.
s e m
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g l as
a

com
m .
m ase
.co


a g l Page 24 of 98

Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

Cut the block in half and stack it. The shape changes; the number of packets does not.

3 × 18 = 54

Double 3 → 6, halve 18 → 9

6 × 9 = 54 ✓

Why it happens: You made one number twice as big and the other half as big. Twice
as many rows, but each row only half as long. The extra and the missing cancel each
other out exactly.

Q2 (b) 22 × 5: fill in ___ × ___ = ____

11 × 10 = 110.

1. Look at the tall block. It is 22 packets tall and 5 packets wide.
2. Cut it across the middle. Each half is 11 tall and 5 wide.
3. Put the two halves side by side. Now the block is 11 tall and 10 wide.
4. Multiply. 11 × 10 = 110 packets.

22 × 5

Step 1. Half of 22 = 11

Step 2. Double of 5 = 10

Step 3. 11 × 10 = 110 packets

Check it yourself: Count the other way. 22 × 5 = 22 × 5. Twenty 5s is 100 and two
more 5s is 10, so 100 + 10 = 110 ✓

Q3 (c) Solve the following problems like the previous ones (fill in the Half and Double
boxes): 14 × 3 = , 38 × 5 = , 16 × 4 = , 35 × 14 =

Halve the first number, double the second, then multiply.

Page 25 of 98

Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

PROBLEM HALF DOUBLE NEW PROBLEM ANSWER

14 × 3 7 6 7×6 42

38 × 5 19 10 19 × 10 190

16 × 4 8 8 8×8 64

35 × 14 7 (half of 14) 70 (double of 35) 70 × 7 490

35 × 14

Step 1. This time halve the second number: half of 14 = 7
Step 2. Double the first: 35 + 35 = 70

Step 3. 70 × 7 = 490

Choose which number to halve: Halve the even number. In 35 × 14, only 14 is even,
so 14 must be the one you halve. In 16 × 4 both are even, so you may choose either.

Check 35 × 14 another way: 35 × 14 = 35 × 10 + 35 × 4 = 350 + 140 = 490 ✓

Q4 This halving and doubling strategy works well when we have to multiply with
numbers like 5 and 25. Discuss why?

Because doubling 5 gives 10, and doubling 25 twice gives 100 — and multiplying by 10 or
100 is the easiest job in arithmetic.

1. Start with 5. Double it once and you get 10. Multiplying by 10 just puts a zero on the end.
2. Start with 25. Double it once and you get 50. Double it again and you get 100. Multiplying
by 100 puts two zeros on the end.
3. Each time you double one number, halve the other. The answer never changes, but the
sum becomes far easier.

Page 26 of 98

Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

Example with 5

18 × 5 → 9 × 10 = 90

Example with 25

16 × 25 → 8 × 50 = 4 × 100 = 400

Why 10 and 100 are so friendly: Our numbers are built in tens. When you multiply
by 10 every digit simply shifts one house to the left. There is nothing to carry and
nothing to remember.

Sample answer for the class discussion: We said 5 is half of 10 and 25 is a quarter
of 100. So whenever 5 or 25 turns up, we can climb up to 10 or 100 by doubling, as
long as we halve the other number the same number of times.

Q5 (d) Find the product by halving and doubling either the multiplier or the
multiplicand. 1) 5 × 18 2) 50 × 28 3) 15 × 22 4) 25 × 12 5) 12 × 45 6) 16 × 45

All six, each shown as a single easy step.

PROBLEM HALVE DOUBLE BECOMES ANSWER

1) 5 × 18 18 → 9 5 → 10 10 × 9 90

2) 50 × 28 28 → 14 50 → 100 100 × 14 1,400

3) 15 × 22 22 → 11 15 → 30 30 × 11 330

4) 25 × 12 12 → 6 25 → 50 50 × 6 300

5) 12 × 45 12 → 6 45 → 90 6 × 90 540

6) 16 × 45 16 → 8 45 → 90 8 × 90 720

Page 27 of 98

Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

2) 50 × 28

Step 1. Half of 28 = 14

Step 2. Double of 50 = 100
Step 3. 100 × 14 = 1,400

4) 25 × 12

Step 1. Half of 12 = 6

Step 2. Double of 25 = 50

Step 3. 50 × 6 = 300

(You could double again: 25 × 12 = 100 × 3 = 300 ✓)

Check one by a second route: 16 × 45 = 16 × 45. Ten 45s is 450 and six 45s is 270.
450 + 270 = 720 ✓

Q6 (e) Give 5 examples of multiplication problems where halving and doubling will help
in finding the product easily. Find the products as well.

Sample answer: Pick problems where one number is even and the other is 5, 25 or 50 — those
are the ones that climb to 10 or 100.

MY EXAMPLE HALVE AND DOUBLE BECOMES PRODUCT

5 × 24 24 → 12, 5 → 10 10 × 12 120

25 × 16 16 → 8, 25 → 50; again 8 → 4, 50 → 100 100 × 4 400

15 × 14 14 → 7, 15 → 30 30 × 7 210

5 × 36 36 → 18, 5 → 10 10 × 18 180

50 × 18 18 → 9, 50 → 100 100 × 9 900

What makes a good example: One number must be even, or you cannot halve it
neatly. The other should be 5, 15, 25 or 50, because doubling those gives 10, 30, 50
or 100.

Page 28 of 98

as e
Class 5 Maths Chapter 6 The Dairy Farm
a g l AglaSem · NCERT Solutions

co m
m.
Try this: A bad example is 7 × 9, because neither number is even and neither is close

m as e
l
to 10 or 100. Halving and doubling would only make it messier. Choosing the right
. conumbers a g
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tool for the is part of the skill.

l a se
ag

co m
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Nearest Multiple — Page 76
m .
Jump to a friendly number, then adjust
as e
NEAREST MULTIPLE a g l

co m
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(a) 4 × 19. Observe the picture and find why we need to subtract 4.

as
Q1

m l
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l a se
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m a s
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l a se
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The picture for 4 × 19 printed on page 76. Each small square is one item.
a

co m
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as e

m l
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Because the picture shows 4 rows of 20 with one column rubbed out, and that column
m
l a se
holds 4 packets.

a g 1. Look at the drawing. It is 4 rows deep and 20 columns wide — that is 4 × 20 = 80 squares.
2. Count the white column at the end. The last column is not filled. It has 4 squares, one from
se m
com g l a
each row.
m . a
ase
agl
3. Take that column away. 80 − 4 = 76.
4. So 4 × 19 = 76.

co m
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4 rows of 20 = 80
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80 − 4 = 76
m . co this empty column holds 4 agl
l a se
ag is twenty columns minus one, and one column here is 4 packets.
Nineteen columns

co m
m .
m ase
.co


a g l Page 29 of 98

Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

4 × 19

Step 1. Go to the nearest easy number: 4 × 20 = 80

Step 2. That is one column of 4 too many.
Step 3. 80 − 4 = 76

Why we subtract 4 and not 1: The column we removed is not one square, it is one
whole column. The block is 4 deep, so the column has 4 squares in it.

Q2 (b) 14 × 21. Observe the picture and find why we need to add 14.

The picture for 14 × 21 printed on page 76. Each small square is one item.

Because 21 columns is 20 columns plus one extra, and that extra column holds 14 packets.

1. Look at the drawing. The yellow part is 14 rows deep and 20 columns wide, so 14 × 20 =
280.
2. Look at the green strip on the right. That is the twenty-first column, coloured green.
3. Count the green squares. The block is 14 deep, so the green column has 14 squares.
4. Add it on. 280 + 14 = 294.

Page 30 of 98

Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

14 × 21

Step 1. 14 × 20 = 280

Step 2. One extra column = 14

Step 3. 280 + 14 = 294

Why the extra column is 14: The number of squares in a column is always the
depth of the block. Here the block is 14 rows deep, so every column — yellow or
green — holds 14 squares.

The rule in one line: Go to the nearest ten or hundred, then add or take away as
many whole groups as you overshot or fell short.

Q3 (c) Give 5 examples of problems where you can use the nearest multiple to find the
product easily. Find the products as well.

Sample answer: choose problems where one number is just below or just above a ten, a
hundred or a nice multiple.

MY EXAMPLE NEAREST MULTIPLE ADJUST PRODUCT

9 × 32 10 × 32 = 320 − 32 288

99 × 7 100 × 7 = 700 −7 693

21 × 15 20 × 15 = 300 + 15 315

49 × 6 50 × 6 = 300 −6 294

101 × 8 100 × 8 = 800 +8 808

What makes a good example: One number should sit within 1 or 2 of a friendly
number like 10, 20, 50, 100. Then you only have to add or take away one or two
groups.

Page 31 of 98

Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

Check one: 49 × 6 = 294. Test it the long way: 40 × 6 = 240 and 9 × 6 = 54, and 240 +
54 = 294 ✓

Q4 (d) Find the products of the following numbers by finding the nearest multiple. 1) 7
× 52 2) 12 × 28 3) 75 × 31 4) 99 × 15 5) 8 × 25 6) 22 × 42

All six, with the friendly number named each time.

PROBLEM NEAREST MULTIPLE ADJUSTMENT ANSWER

1) 7 × 52 7 × 50 = 350 + 7 × 2 = 14 364

2) 12 × 28 12 × 30 = 360 − 12 × 2 = 24 336

3) 75 × 31 75 × 30 = 2,250 + 75 2,325

4) 99 × 15 100 × 15 = 1,500 − 15 1,485

5) 8 × 25 10 × 25 = 250 − 2 × 25 = 50 200

6) 22 × 42 22 × 40 = 880 + 22 × 2 = 44 924

3) 75 × 31

Step 1. 31 is one more than 30.

Step 2. 75 × 30 = 75 × 3 × 10 = 225 × 10 = 2,250
Step 3. Add one more group of 75.

Step 4. 2,250 + 75 = 2,325

6) 22 × 42

Step 1. 42 is two more than 40.

Step 2. 22 × 40 = 22 × 4 × 10 = 88 × 10 = 880

Step 3. Two more groups of 22 = 44

Step 4. 880 + 44 = 924

Page 32 of 98

Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

Check 12 × 28 a second way: 12 × 28 = 12 × 20 + 12 × 8 = 240 + 96 = 336 ✓

Let Us Solve — Page 77
Word problems: use the strategies flexibly

LET US SOLVE

Q1 A school has an auditorium with 35 rows, with 42 seats in each row. How many
people can sit in this auditorium?

1,470 people can sit in the auditorium.
We need the total number of seats. There are 35 rows, and each row holds 42 seats.

Total seats = 35 × 42
Step 1. Split 42 into 40 and 2.

Step 2. 35 × 40 = 35 × 4 × 10 = 140 × 10 = 1,400

Step 3. 35 × 2 = 70

Step 4. 1,400 + 70 = 1,470 seats

Check it yourself with doubling and halving: 35 × 42 = 70 × 21. And 70 × 21 = 70 ×
20 + 70 = 1,400 + 70 = 1,470 ✓

Q2 Priya jogs 4 kilometres every day. How many kilometers will she jog in 31 days?

Priya will jog 124 kilometres in 31 days.
We need the total distance. She covers 4 km on each of 31 days.

Page 33 of 98

ase
Class 5 Maths Chapter 6 The Dairy Farm
a g l AglaSem · NCERT Solutions

co m
e m.
Total distance = 4 × 31
m l as
.co
Step 1. Go to the nearest easy number: 4 × 30 = 120
m a g
l a se
g
Step 2. One more day means 4 more kilometres.
aStep 3. 120 + 4 = 124 kilometres

co m
m . ag
l a se
Did you know? 31 days is one full month like January or March. So Priya jogs about
ag the distance from Delhi to Agra.
124 km in such a month — roughly

co m
em.
m l as
.co a g
Q3 A school has received 36 boxes of books with 48 books in each box. How many total

a s em books did the school receive in the boxes?
l
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m a s
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The school received 1,728 books.

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We need the total number of books. There are 36 boxes with 48 books in each.

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Step 1. Split 48 into 40 and 8.
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m 3. 36 × 8 = 288 a g
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Step 4. 1,440 + 288 = 1,728 books
a se
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Check it yourself with theanearest
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a 36 × 2 = 72. So 1,800 − 72 = 1,728 ✓
many, and two boxes are

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Priya uses 16 metres of cloth to make 4 kurtas. How much

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agl
Notice that 8 kurtas is exactly double

co m
m .
m ase
.co


a g l Page 34 of 98

Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

Step 1. 8 kurtas = 2 × 4 kurtas

Step 2. Cloth needed = 2 × 16 m

Step 3. 2 × 16 = 32 metres

Another way, if you prefer to find one kurta first:

1. Cloth for one kurta = 16 m ÷ 4 = 4 m.
2. Cloth for eight kurtas = 4 m × 8 = 32 m.

Why doubling works here: Every kurta uses the same amount of cloth. If you sew
twice as many kurtas, you must buy twice as much cloth. The relationship stays in
step.

Q5 Gollappa has 29 cows on his farm. Each cow produces 5 litres of milk per day. How
many litres of milk do the cow produce in total, each day?

The cows produce 145 litres of milk each day.
We need the total milk in one day. There are 29 cows, and each gives 5 litres.

Total milk = 29 × 5

Step 1. Go to the nearest ten: 30 × 5 = 150

Step 2. That counts one cow too many, so take away 5 litres.

Step 3. 150 − 5 = 145 litres

Check it yourself: 20 × 5 = 100 and 9 × 5 = 45. Then 100 + 45 = 145 litres ✓

Q6 Maska Cow Farm has 297 cows. Each cow requires 18 kg of fodder per day. How
much total fodder is needed to feed 297 cows every day?

5,346 kg of fodder is needed every day.
Fodder is the grass and grain a cow eats. We need the total for 297 cows, each eating 18 kg.

Page 35 of 98

Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

Total fodder = 297 × 18

Step 1. 297 is close to 300, so start there.

Step 2. 300 × 18 = 3 × 18 × 100 = 54 × 100 = 5,400
Step 3. We counted 3 cows too many. 3 × 18 = 54

Step 4. 5,400 − 54 = 5,346 kg

Check it yourself the long way: 297 × 10 = 2,970 and 297 × 8 = 2,376. Then 2,970 +
2,376 = 5,346 kg ✓

Waste and Composting — Pages 77–78
Nida's, Kanti's and John's ways of multiplying

WASTE AND COMPOSTING

Q1 A family of 4 produces around 35 kg of kitchen waste in a month. How much waste
will the family produce in a year? The family produces _____ kg of waste in a year.

The family produces 420 kg of kitchen waste in a year.
A year has 12 months. Each month brings 35 kg of waste.

Waste in a year = 12 × 35

Step 1. Split 35 into 30 and 5, and 12 into 10 and 2.

Step 2. 10 × 30 = 300

Step 3. 10 × 5 = 50

Step 4. 2 × 30 = 60
Step 5. 2 × 5 = 10

Step 6. 300 + 50 + 60 + 10 = 420 kg

Nida's grid shows the same four pieces in a box:

Page 36 of 98

Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

× 30 KG 5 KG

10 300 50

2 60 10

Column totals 360 60

Grand total 420 kg

Why the grid has four boxes: 12 was split into 2 parts and 35 was split into 2 parts.
Two parts times two parts makes four small rectangles. Add all four and you get the
whole rectangle back.

Did you know? Kitchen waste — vegetable peels, tea leaves, leftover rice — does
not have to go to the dump. Buried in a pit it turns into compost, a rich brown food
for plants.

Q2 Kanti and John tried to solve it in the following ways. How are these solutions same
or different? Discuss in class.

Both are the same idea; they only differ in how many lines they write. Both split 35 into 30 +
5 and 12 into 10 + 2, and both reach 420.

KANTI WRITES FOUR LINES JOHN WRITES TWO LINES

10 (2 × 5) 60 + 10 = 70 (2 × 35)

60 (2 × 30)

50 (10 × 5) 300 + 50 = 350 (10 × 35)

300 (10 × 30)

Total 420 Total 420

1. Same pieces. Both use 10, 60, 50 and 300. Not one piece is missing from either.
2. Different grouping. Kanti keeps all four pieces apart. John joins the two pieces that came
from the same row: he adds 60 + 10 straight away to make 70, and 300 + 50 to make 350.
3. Same answer. Kanti: 10 + 60 + 50 + 300 = 420. John: 70 + 350 = 420.

Page 37 of 98

Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

Which is better? Kanti's way shows every step, so it is easier to understand and
easier to check. John's way is shorter, so it is faster once you are sure of yourself.
Both are correct — pick the one you can do without slipping.

Sample answer for the class discussion: We said John's method is really Kanti's
method with the pairs already added up. Nida's grid is the same again, drawn as a
picture instead of a list. Three faces of one idea: break the numbers into place
values, multiply the pieces, add them all back.

Q3 The family regularly composts the collected waste. They get around 150 kg of
compost each year, which they use in their garden. 1 kg of compost is often sold in
the market at ₹24. By creating their own compost, how much money have they
saved in a year?

The family saves ₹3,600 in a year.
They did not have to buy 150 kg of compost. Each kilogram would have cost ₹24.

Money saved = 24 × 150

Step 1. Split 150 into 100 + 50, and 24 into 20 + 4.

Step 2. 20 × 100 = 2,000

Step 3. 20 × 50 = 1,000

Step 4. 4 × 100 = 400

Step 5. 4 × 50 = 200

Step 6. 2,000 + 1,000 + 400 + 200 = ₹3,600

× 100 50

20 2,000 1,000

4 400 200

Column totals 2,400 1,200

Grand total ₹3,600

Page 38 of 98

as e
Class 5 Maths Chapter 6 The Dairy Farm
a g l AglaSem · NCERT Solutions

co m
m.
Check it a shorter way: 24 × 150 = 24 × 100 + 24 × 50 = 2,400 + 1,200 = ₹3,600 ✓

m as e
.co a g l
s em
Why this counts as saving: Money saved is money you did not have to spend. By
a
gl their own kitchen waste into compost, the family kept ₹3,600 in their pocket
aturning
— and kept 420 kg of waste out of the rubbish heap.

com
e m . ag
g l as
Let Us Multiply — Page 79 a

co m
em.
m l as
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m ase
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a g l Page 39 of 98

Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

Filling the grid and the column methods

LET US MULTIPLY

Q1 (a) 32 × 8. Fill the grid and complete the two column methods.

× 30 2

8

32 (30 + 2)
× 8

____ (8 × 2)
__0 (8 × 30)

256

Page 40 of 98

Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

32 (30 + 2)
× 8

____0 + 16 = 256

The grid and the two column methods printed on page 79. Some numbers are already
filled in.

32 × 8 = 256. Here is every blank filled in.

× 30 2

8 240 16

Total 240 + 16 = 256

Column method, all the pieces

32 (30 + 2)

×8

Step 1. 8 × 2 = 16

Step 2. 8 × 30 = 240

Step 3. 16 + 240 = 256

Page 41 of 98

Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

Column method, joined up

32 (30 + 2)

×8
240 + 16 = 256

Why 8 × 30 is 240 and not 24: The 3 in 32 is not three, it is three tens. So 8 × 3 tens =
24 tens = 240. Losing that zero is the commonest slip in the whole chapter.

Check it yourself: 32 × 8 = 32 × 4 × 2 = 128 × 2 = 256 ✓

Page 42 of 98

Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

Q2 (b) 69 × 45. Fill the grid and complete the column methods.

× 60 9

40

5

2700 405

3105

69 (60 + 9)
× 45 (40 + 5)

45 (5 × 9)
___0 (5 × 60)
___0 (40 × 9)
__00 (40 × 60)

3,105

Page 43 of 98

as e
Class 5 Maths Chapter 6 The Dairy Farm
a g l AglaSem · NCERT Solutions

. com
m 69 (60 as+em9)
. co a gl
e m
g l as × 45 (40 + 5)
a
300 + .__ com = 345 (5 × 69) ag
a s em
____ +agl___ = 2760 (40 × 69)

. com
m a em
= s3,105
e m . co agl
g l as
a
The grid and the two column methods printed on page 79. Some numbers are already
filled in.

m a s
m .co agl
l a se
a g
69 × 45 = 3,105. Every blank filled in.

com
m .
as e
× 60 9 ROW TOTAL

40 .co
m g l
m a
ase
2,400 360 2,760

agl 5 300 45 345

se m
Column total 2,700
com
405 3,105
g l a
m . a
ase
Column method, all four pieces agl
m
69 (60 + 9) × 45 (40 + 5)
. co
e m
as
Step 1. 5 × 9 = 45

com2. 5 × 60 = 300
.Step a g l
se m
g l a
a
Step 3. 40 × 9 = 360
.c
Step 4. 40 × 60 = 2,400
s e m
m a
Step 5. 45 + 300 + 360 + 2,400 = 3,105
e m . co agl
g l as
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Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

Column method, two rows

Step 1. 300 + 45 = 345 (5 × 69)

Step 2. 2,400 + 360 = 2,760 (40 × 69)

Step 3. 345 + 2,760 = 3,105

Read the grid two ways: Adding across the rows gives 345 and 2,760 — that is
John's method. Adding down the columns gives 2,700 and 405 — the totals printed
in the book. Both add up to 3,105, because you are adding the same four boxes.

Check it yourself: 69 × 45 = 69 × 45. Try the nearest multiple: 70 × 45 = 3,150, then
take away one 45. 3,150 − 45 = 3,105 ✓

Let Us Do — Pages 79–80
Solve like Nida, like Kanti, like John; then word problems

LET US DO

Q1 Solve the following problems like Nida did (using a grid). a) 78 × 4 b) 83 × 9 c) 67 × 28
d) 53 × 37

Nida's way is to draw a grid, split each number into its place values, multiply every box, then
add.
a) 78 × 4 = 312

× 70 8

4 280 32

Total 280 + 32 = 312

b) 83 × 9 = 747

× 80 3

9 720 27

Total 720 + 27 = 747

Page 45 of 98

Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

c) 67 × 28 = 1,876

× 60 7 ROW TOTAL

20 1,200 140 1,340

8 480 56 536

Total 1,340 + 536 = 1,876

d) 53 × 37 = 1,961

× 50 3 ROW TOTAL

30 1,500 90 1,590

7 350 21 371

Total 1,590 + 371 = 1,961

How to fill a grid

Step 1. Split each number into its place values (67 → 60 + 7).

Step 2. Write one number across the top and the other down the side.

Step 3. Multiply each pair and write the answer in that box.

Step 4. Add every box.

Check c) another way: 67 × 28 = 67 × 30 − 67 × 2 = 2,010 − 134 = 1,876 ✓

Q2 Solve the following problems like Kanti. a) 94 × 5 b) 49 × 6 c) 37 × 53 d) 28 × 79

Kanti's way is to write every small piece on its own line, smallest first, then add them all.

Page 46 of 98

Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

a) 94 × 5

94 (90 + 4) × 5

Step 1. 5 × 4 = 20

Step 2. 5 × 90 = 450

Step 3. 20 + 450 = 470

b) 49 × 6

49 (40 + 9) × 6

Step 1. 6 × 9 = 54

Step 2. 6 × 40 = 240

Step 3. 54 + 240 = 294

c) 37 × 53

37 (30 + 7) × 53 (50 + 3)

Step 1. 3 × 7 = 21

Step 2. 3 × 30 = 90

Step 3. 50 × 7 = 350

Step 4. 50 × 30 = 1,500

Step 5. 21 + 90 + 350 + 1,500 = 1,961

d) 28 × 79

28 (20 + 8) × 79 (70 + 9)

Step 1. 9 × 8 = 72

Step 2. 9 × 20 = 180

Step 3. 70 × 8 = 560

Step 4. 70 × 20 = 1,400

Step 5. 72 + 180 + 560 + 1,400 = 2,212

Page 47 of 98

Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

Why four lines for c) and d): Both numbers were split into two parts, so there are 2
× 2 = 4 pieces to multiply. For a) and b) only one number was split, so there are just 2
pieces.

Check d): 28 × 79 = 28 × 80 − 28 = 2,240 − 28 = 2,212 ✓

Q3 Solve the following problems like John. a) 86 × 3 b) 72 × 7 c) 94 × 36 d) 66 × 22

John's way is one line for each digit of the second number, with the pieces of that line already
added together.

a) 86 × 3

Step 1. 3 × 86 → 240 + 18 = 258

b) 72 × 7

Step 1. 7 × 72 → 490 + 14 = 504

c) 94 × 36

Step 1. 6 × 94 → 540 + 24 = 564

Step 2. 30 × 94 → 2,700 + 120 = 2,820

Step 3. 564 + 2,820 = 3,384

d) 66 × 22

Step 1. 2 × 66 → 120 + 12 = 132

Step 2. 20 × 66 → 1,200 + 120 = 1,320

Step 3. 132 + 1,320 = 1,452

Watch the second line. In c) the 3 of 36 is really 30, so the line is 30 × 94, not 3 × 94.
That is why 2,820 is ten times bigger than 282.

Page 48 of 98

as e
Class 5 Maths Chapter 6 The Dairy Farm
a g l AglaSem · NCERT Solutions

co m
m.
Check d) with doubling and halving: 66 × 22 = 33 × 44. And 33 × 44 = 33 × 40 + 33 ×

m as e
l
4 = 1,320 + 132 = 1,452 ✓

m .co a g
l a se
a g
Q4 (a) A movie theater has 8 rows of seats, and each row has 12 seats. If half the seats

co m
ag
are filled, how many people are watching the movie? If 3 more rows get filled, how

m .
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many total people will be there?

g l as
ANSWER a
48 people are watching. After 3 more rows fill up, there will be 84 people.
co m
em.
as
First find how many seats the whole theatre has.
m l
.co a g
a s em1. Total seats = 8 × 12 = 96 seats
gl
Step
a Step 2. Half the seats = 96 ÷ 2 = 48 people

m a s
.co agl
Step 3. Seats in 3 more rows = 3 × 12 = 36 seats

se m
Step 4. New total = 48 + 36 = 84 people
g l a
a
Why 3 rows is 36 people: Every row holds 12 seats. Three full rows means three lots
co m
m .
of 12, which is 36.
m as e
.co a g l
s m it yourself: 84 people out of 96 seats leaves 96 − 84 = 12 seats empty —
eCheck
gl a
a exactly one row. That fits: half of 8 rows is 4 rows, plus 3 more rows makes 7 rows full

se m
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and 1 row empty ✓
. a g l
m
gl ase
a
(b) In a test match between India and West Indies, the Indian team hit twenty-four
Q5

co
4s and eighteen 6s across the two innings. How many runs were scored in 4s and 6s
m
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each? 234 runs were made by running between the wickets. If 23 runs were extras,

m l as
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how many runs were scored by Indian team in the two innings?

emANSWER a
a s
agl
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Runs in 4s = 96. Runs in 6s = 108. Total runs in the two innings = 461.

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Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

Runs from fours

Step 1. 24 fours = 24 × 4

Step 2. 24 × 4 = 96 runs

Runs from sixes

Step 1. 18 sixes = 18 × 6

Step 2. 18 × 6 = 108 runs

Total for the team

Step 1. Boundaries = 96 + 108 = 204 runs

Step 2. Add the running between wickets: 204 + 234 = 438 runs

Step 3. Add the extras: 438 + 23 = 461 runs

Why we add all three: A team's score is made of every run it gets — runs hit to the
boundary, runs made by running, and extras given away by the other team. Nothing
is counted twice, because a four is never also a run between the wickets.

Did you know? A four means the ball reaches the boundary rope along the ground.
A six means it clears the rope in the air. Eighteen sixes across a test match is a big-
hitting side.

Q6 (c) Anjali buys 15 bulbs and 12 tube lights from Sudha Electricals. Each bulb costs
₹25 and each tube light costs ₹34. How much money should Anjali give to the
shopkeeper?

Anjali should give ₹783 to the shopkeeper.
There are two different things being bought, so find each cost and then add.

Page 50 of 98

Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

Cost of bulbs

Step 1. 15 bulbs × ₹25 each

Step 2. Use doubling and halving: 15 × 25 = 30 × 12.5 — messy. Try place values instead.

Step 3. 15 × 25 = 10 × 25 + 5 × 25 = 250 + 125 = ₹375

Cost of tube lights

Step 1. 12 tube lights × ₹34 each

Step 2. 12 × 34 = 10 × 34 + 2 × 34

Step 3. 340 + 68 = ₹408

Total to pay

₹375 + ₹408 = ₹783

Check the bulbs a second way: 15 × 25 = 15 × 100 ÷ 4 = 1,500 ÷ 4 = 375 ✓ (because
25 is one quarter of 100).

Q7 (d) A shopkeeper sold 28 bags of rice. Each bag costs ₹350. How much money did he
earn by selling rice bags?

He earned ₹9,800.

Money earned = 28 × 350

Step 1. Set the zero of 350 aside: 28 × 35

Step 2. 28 × 35 = 28 × 30 + 28 × 5

Step 3. 28 × 30 = 840

Step 4. 28 × 5 = 140
Step 5. 840 + 140 = 980

Step 6. Put the zero back: ₹9,800

Page 51 of 98

Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

Check it with doubling and halving: 28 × 350 = 14 × 700 = 14 × 7 × 100 = 98 × 100 =
₹9,800 ✓

Q8 (e) A school library has 86 shelves and each shelf has 162 books. Find the number of
books in the library.

The library has 13,932 books.

Total books = 86 × 162

Step 1. Split 162 into 100 + 60 + 2.

Step 2. 86 × 100 = 8,600

Step 3. 86 × 60 = 5,160

Step 4. 86 × 2 = 172

Step 5. 8,600 + 5,160 + 172 = 13,932 books

Why 86 × 60 is 5,160: 86 × 6 = 516. Then 60 is ten times 6, so the answer is ten times
516, which is 5,160.

Check it yourself: Estimate first. 86 is close to 90 and 162 is close to 160, so the
answer should be near 90 × 160 = 14,400. Our answer 13,932 is close to that, so it
looks right ✓

Dairy Cooperative — Pages 81–83
Mili's father's method: multiplying with place values

DAIRY COOPERATIVE MATH TALK

Q1 A dairy cooperative in a small town gets its milk from 268 villagers. Each villager
has at least 4 milk-giving cows. What is the minimum number of cows the dairy
cooperative gets its milk from?

The dairy cooperative gets milk from at least 1,072 cows.

Page 52 of 98

Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

Each of the 268 villagers has at least 4 cows. So the smallest possible number of cows is 268 × 4.

Minimum cows = 268 × 4

Step 1. Split 268 into 200 + 60 + 8.
Step 2. 4 × 200 = 800

Step 3. 4 × 60 = 240

Step 4. 4 × 8 = 32

Step 5. 800 + 240 + 32 = 1,072 cows

Why "at least" matters: Some villagers may own 5 or 6 cows. But nobody owns
fewer than 4. So 4 cows each gives the smallest total the cooperative could possibly
have — the minimum.

Did you know? A dairy cooperative is a group of villagers who pool their milk and
sell it together, so each family gets a fair price. Many are run by women, like
Jamanaben Maganbhai Naku of Tuked village in Surat, who keeps 27 Gir cows.

Q2 Discuss how this multiplication is being carried out! (Mili's father's method for 268 ×
4, with 30 added to 4 × 60 and 200 added to 4 × 200)

In Mili's father's method you multiply one column at a time, from the right, and any extra
tens or hundreds are carried to the next column — where they are added to the product,
not to the original digit.

1. Ones column. 4 × 8 = 32. That is 3 tens and 2 ones. Write 2 in the ones place and carry the 3
tens to the tens column.
2. Tens column. 4 × 6 tens = 24 tens = 240. Now add the carried 3 tens: 240 + 30 = 270. Write 7
in the tens place and carry 2 hundreds.
3. Hundreds column. 4 × 2 hundreds = 8 hundreds = 800. Add the carried 200: 800 + 200 =
1,000. Write 0 in the hundreds place and 1 in the thousands place.
4. Read the answer. 1,072.

Page 53 of 98

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Class 5 Maths Chapter 6 The Dairy Farm
a g l AglaSem · NCERT Solutions

co m
Oas e m.
m Th H T
l
m .co a g
l a se 2 3
a g
2 6 8
co m
×
e m . 4 ag
g l as
1 a 0 7 2
om
The circled digits are the carries: 3 tens and 2 hundreds.
. c
e m
c o m The carried 3 joins 4 × 60, not 60. The carried 2 joins 4 × 200,
g l as 200.
not

m . a
l a se
a g The point to notice: The carried 3 is added after the multiplication of that column,

s
not before. If you added 3 to the 6 first you would be working out 4 × 90, which is
m a
completely wrong. Multiply first, then add the carry.
m .co agl
l a se
a g
Sample answer for the class discussion: We said each column is multiplied on its

m
own, and whatever spills over goes next door. It is like filling small boxes of 10: when
. co
m
a box is full it moves up to the next shelf.

m as e
.co a g l
se m
g l a
a Q3 The villagers rear various breeds of cows. Among the villagers, there are 453 Gir
cows. Each of these cows give 13 litres of milk each day. How many litres of milk
se m
com g l a
.
does the dairy cooperative receive from Gir cows everyday?
m a
ase
agl
The cooperative receives 5,889 litres of milk from Gir cows every day.

co m
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Each Gir cow gives 13 litres. There are 453 of them.

em
m l as
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Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

Milk each day = 453 × 13

Step 1. Multiply by the 3 first.

3 × 400 = 1,200

3 × 50 = 150

3×3=9

Step 2. Add them: 1,200 + 150 + 9 = 1,359

Step 3. Now multiply by the 10.

10 × 400 = 4,000

10 × 50 = 500

10 × 3 = 30

Step 4. Add them: 4,000 + 500 + 30 = 4,530

Step 5. 1,359 + 4,530 = 5,889 litres

Why the second row ends in 0: The 1 of 13 stands for 1 ten. So 1 ten × 453 = 453
tens = 4,530. Forgetting that zero would make the answer ten times too small.

Did you know? The Gir is an Indian breed of cow from Gujarat. It is famous for
giving a lot of milk and for coping well with heat.

Q4 The dairy cooperative sells 1 kg cow ghee for ₹574. The cooperative is able to
produce around 125 kg of ghee in a month. How much do they earn by selling ghee
in a month?

The cooperative earns ₹71,750 in a month.
They sell 125 kg of ghee, and each kilogram brings ₹574.

Page 55 of 98

Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

Earnings = 574 × 125

Row 1 — multiply by 5
5 × 500 = 2,500

5 × 70 = 350

5 × 4 = 20

2,500 + 350 + 20 = 2,870

Row 2 — multiply by 20

20 × 500 = 10,000

20 × 70 = 1,400
20 × 4 = 80

10,000 + 1,400 + 80 = 11,480

Row 3 — multiply by 100

100 × 500 = 50,000

100 × 70 = 7,000

100 × 4 = 400

50,000 + 7,000 + 400 = 57,400

Add the three rows

2,870 + 11,480 + 57,400 = ₹71,750

Check it a quicker way: 125 is one eighth of 1,000. So 574 × 125 = 574 × 1,000 ÷ 8 =
574,000 ÷ 8 = 71,750 ✓

Why there are three rows: 125 has three digits — 100, 20 and 5. Each digit gets its
own row, and each row must carry the right number of zeros.

Page 56 of 98

Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

Q5 Discuss what the same coloured digits might mean. (in the 574 × 125 working)

The digits printed in the same colour are a carry and the place it lands in. A coloured digit
above the line is the amount that spilled over from the column to its right.

1. Every column can overflow. When a column's product is 10 or more, only the last digit
stays. The rest must move one house to the left.
2. The colour ties the two together. One coloured digit is the little number written above the
next column. The matching coloured digit is the part of the product it came from.
3. Example from the working. 2 tens × 574 gives 11,480. The 1 that appears above the
thousands column is a carry, and it is the same colour as the digit that produced it.
4. Reminder. A carry is always added after the next multiplication, never before it.

Why colours help: In a long multiplication there can be five or six carries floating
about. Colouring them shows a learner where each one came from and where it
must go, so no carry gets lost or used twice.

Sample answer for the class discussion: We said the colours are like labels on
parcels. Each parcel of tens or hundreds has to be delivered to the correct column,
and the colour tells you the address.

Let Us Solve — Pages 83–84
Practise Nida's, John's and Mili's father's methods

LET US SOLVE

Q1 Solve the following problems like Nida. a) 548 × 6 b) 682 × 3 c) 324 × 18 d) 507 × 23 e)
190 × 65

Nida's grid: split both numbers into place values, multiply every box, add everything.
a) 548 × 6 = 3,288

× 500 40 8

6 3,000 240 48

Page 57 of 98

Class 5 Maths Chapter 6 The Dairy Farm AglaSem · NCERT Solutions

3,000 + 240 + 48 = 3,288

b) 682 × 3 = 2,046

× 600 80 2

3 1,800 240 6

1,800 + 240 + 6 = 2,046

c) 324 × 18 = 5,832

× 300 20 4 ROW TOTAL

10 3,000 200 40 3,240

8 2,400 160 32 2,592

3,240 + 2,592 = 5,832

d) 507 × 23 = 11,661

× 500 0 7 ROW TOTAL

20 10,000 0 140 10,140

3 1,500 0 21 1,521

10,140 + 1,521 = 11,661

e) 190 × 65 = 12,350

× 100 90 ROW TOTAL

60 6,000 5,400 11,400

5 500 450 950

11,400 + 950 = 12,350

Page 58 of 98

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Class 5 Maths Chapter 6 The Dairy Farm
a g l AglaSem · NCERT Solutions

co m
m.
Do not skip the zero. In d) the middle digit of 507 is 0, so those two boxes are 0.

as e
om to remind yourself. l
Many students forget the tens column altogether and lose a place — keep the empty
box in the.cgrid a g
a s em
agl

co m
ag
Solve the following problems like John. a) 123 × 84 b) 368 × 32 c) 159 × 324 d) 239 ×
.
Q2

e m
as
401 e) 592 × 5 f) 101 × 22

a g l

John's way: one line for each digit of the multiplier, with that line's pieces already added.
co m
em.
m= 10,332
a) 123 .×co84 g l as
a
sem1. 4 × 123 → 400 + 80 + 12 = 492
aStep
agl
Step 2. 80 × 123 → 8,000 + 1,600 + 240 = 9,840
m a s
Step 3. 492 + 9,840 = 10,332
m .co agl
l a se
a g
b) 368 × 32 = 11,776

co m
Step 1. 2 × 368 → 600 + 120 + 16 = 736
m .
m as e
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Step 2. 30 × 368 → 9,000 + 1,800 + 240 = 11,040
a g l
s m 3. 736 + 11,040 = 11,776
eStep
gl a
a
se m
com g l a
c) 159 × 324 = 51,516
m . a
e
+ s36 = 636
Step 1. 4 × 159 → 400 + 200la
a g
Step 2. 20 × 159 → 2,000 + 1,000 + 180 = 3,180

co m
Step 3. 300 × 159 → 30,000 + 15,000 + 2,700 = 47,700
m .
m as e
.co
Step 4. 636 + 3,180 + 47,700 = 51,516
a g l
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a c
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