Class 5 Maths Chapter 6 Solutions – The Dairy Farm (NCERT)

NCERT Textbook Pages 70-71

Let Us Think

Question 1.
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.

Solution:

This problem can be solved using the trial-and-error method.
One possible set of values for the shapes is given below.

So, 24 × 3 = 6 × 12 = 72

Question 2.

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

Solution:

When the product 106 is rounded to the nearest ten, it becomes 110.

This value is closer to 100 than any of the other products that can be formed using the given digits.

Question 3.

A dairy has packed butter milk pouches in the following manner. Find the number of pouches kept in each arrangement. One is done for you.

Solution:

Question 4.

Which number am I?

I am a two-digit number, help of the following clues.

(а) 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.

Solution:

Step 1:
List all two-digit multiples of 9 that are less than 50: 18, 27, 36, 45.

Step 2:
Eliminate numbers that are multiples of 4. Since 36 is divisible by 4, remove it.
The remaining numbers are 18, 27, 45.

Step 3:
Keep only odd numbers. As 18 is even, discard it.
The numbers left are 27 and 45.

Step 4:
Exclude multiples of 11 such as 22, 33, 44, 55, ….
Neither 27 nor 45 is a multiple of 11, so both remain.

Step 5:
Both 27 and 45 are less than 50, so they satisfy this condition.

Step 6:
According to the clue, the ones digit must be even.
However, 27 ends in 7 and 45 ends in 5, both of which are odd.
This creates a contradiction.

Conclusion:
No two-digit number satisfies all the given conditions.

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

Solution:

All the clues except (a) and (b) were useful in attempting to determine the number.

If clue (d) is ignored, the possible numbers are 18 or 36.

Similarly, if clues (g) and (h) are omitted and only clue (e) is considered, the number could be 27 or 45.

Question 5.

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?

Solution:

For example, 36 − 25 = 11 (answers might be different), or 36 ÷ 4 = 9, and so on.

Which numbers could you not make? Is it possible to make these numbers using three numbers? You can use two operations, if needed. Which numbers between 0-20 can you get in more than one way?

Solution:
Students should do it by themselves.

NCERT Textbook Pages 72-74

Order of Numbers in Multiplication

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.

Solution:

What is 9 × 0? 0 × 9?

Solution:
9 × 0 = 0 and 0 × 9 = 0

Is this true for the product of any two numbers. Discuss in class.

Solution:

In each case, the number of groups and the size of each group are swapped, yet the total number of butter packets stays unchanged.

Patterns in Multiplication by 10s and 100s

Question 1.
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 = _____ = ______

Solution:
(a) 4 × 10 = 40
(b) 20 × 10 = 200
(c) 10 × 40 = 400
(d) 10 × 10 = 100
(e) 20 × 50 = 1000
(f) 80 × 10 = 800
(g) 3 × 100 = 100 × 3 = 300
(h) 8 × 100 = 100 × 8 = 800
(j) 10 × 100 = 100 × 10 = 1000

Question 2.
Find the answers to the following questions. Fill in the table below and describe the pattern. Discuss in class.

Solution:
(100 × 90 = 9,000)
( 400 × 10 = 4.000)
(60 × 50 = 3.000)
(30 × 20 = 600)
(700 × 4 = 2,800)
(10 × 45 = 450)

How should we write 450 in the table below?

Solution:

Solution:

Let us fill in the table and observe the patterns.

Solution:

NCERT Textbook Pages 75-76

Doubling and Halving

Butter packets are arranged in the following ways. Let us find some strategies to calculate the total number of packets.   

Solution: Given multiplication: 22 × 5

Apply the same strategy:

• Halve 22 → 11
  
  

• Double 5 → 10
  
  

So,
22 × 5 = 11 × 10 = 110

Final Answer

22×5=11×10=11022 \times 5 = 11 \times 10 = 11022×5=11×10=110

(c) Solve the following problems like the previous ones.

Solution:

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

Solution:

No, for numbers such as 5 and 25, the same method cannot be applied directly. Since these numbers are odd, halving them is not convenient and would make the calculation more difficult. Therefore, a different strategy needs to be used in such cases.

(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

Solution:

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

Solution:

NCERT Textbook Page 76

Nearest Multiple

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

Solution:

(1) Multiply 5 by 31 by breaking 31 into 30 and 1:
5 × 30 = 150, and adding 5 gives 155.

(2) To find 7 × 29, multiply 7 by 30 and then subtract 7:
210 − 7 = 203.

(3) For 12 × 49, first calculate 12 × 50 and then subtract 12:
600 − 12 = 588.

(4) In 8 × 101, multiply 8 by 100 and add 8:
800 + 8 = 808.

(5) To solve 16 × 99, multiply 16 by 100 and subtract 16:
1600 − 16 = 1584.

(Other valid methods and answers are also possible.)

(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

Solution:

(1) To calculate 7 × 52, split 52 into 50 and 2:
7 × 50 = 350 and 7 × 2 = 14.
Adding them gives 364.

(2) For 12 × 28, multiply 12 by 30 and subtract the product of 12 and 2:
360 − 24 = 336.

(3) In 75 × 31, first find 75 × 30 and then add 75:
2250 + 75 = 2325.

(4) To solve 99 × 15, multiply 15 by 100 and subtract 15:
1500 − 15 = 1485.

(5) For 8 × 25, multiply 25 by 10 and subtract twice 25:
250 − 50 = 200.

(6) To calculate 22 × 42, break 42 into 40 and 2:
22 × 40 = 880 and 22 × 2 = 44.

Adding them gives 924.

NCERT Textbook Page 77

Let Us Solve

Use strategies flexibly to answer the following questions. Discuss you thoughts in class.

Question 1.
A school has an auditorium with 35 rows, with 42 seats in each row. How many people can sit in this auditorium?

Solution:

There are 35 rows in the auditorium, with 42 seats in each row.
The total number of seats is found by multiplying the two:

35 × 42 = (35 × 40) + (35 × 2)
= 1400 + 70
= 1470

Therefore, the auditorium can accommodate 1470 people.

Question 2.

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

Solution:

Priya jogs 4 km each day.
Over a period of 31 days, the total distance she covers is:

4 × 31 = (4 × 30) + 4
= 120 + 4
= 124 km

Hence, Priya will jog a total of 124 km in 31 days.

Question 3.

A school has received 36 boxes of books with 48 books in each box. How many total books did the school receive in the boxes?

Solution:

Each box contains 48 books, and the school receives 36 boxes.
The total number of books can be calculated as follows:

36 × 48 = (36 × 50) − (36 × 2)
= 1800 − 72
= 1728

Therefore, the school received 1728 books in total.

Question 4.

Priya uses 16 metres of cloth to make 4 kurtas. How much cloth would she need to make 8 kurtas?

Solution:

The cloth required to make 4 kurtas is 16 metres, which means each kurta uses 4 metres of cloth.

To make 8 kurtas, the total cloth needed is calculated as:
4 × 8 = (4 × 10) − (4 × 2)
= 40 − 8
= 32 metres

Therefore, Priya needs 32 metres of cloth to stitch 8 kurtas.

Question 5.

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?

Solution:

Gollappa has 29 cows on his farm, and each cow gives 5 litres of milk.
The total milk produced in a day is calculated as:

5 × 29 = (5 × 30) − 5
= 150 − 5
= 145 litres

Therefore, the farm produces 145 litres of milk per day.

Question 6.

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?

Solution:

There are 297 cows on Maska’s farm, and each cow requires 18 kg of fodder per day.

The total fodder needed daily is calculated as:

18 × 297 = (18 × 300) − (18 × 3)
= 5400 − 54
= 5346 kg

Hence, 5346 kg of fodder is required each day for 297 cows.

NCERT Textbook Pages 79

Let Us Multiply

(a) 32 × 8

Solution:

(b) 69 × 45

Solution:

NCERT Textbook Pages 79-80

Let Us Do

Question 1.
Solve the following problems like Nida did.

(a) 78 × 4

Solution:

(b) 83 × 9

Solution:

(c) 67 × 28

Solution:

(d) 53 × 37

Solution:

Question 2.

Solve the following problems like Kanti did.

a) 94 × 5

Solution:

b) 49 × 6

Solution:

c) 37 × 53

Solution:

d) 28 × 79

Solution:

Question 3.

Solve the following problems like John.

(a) 86 × 3

Solution:

(b) 72 × 7

Solution:

(c) 94 × 36

Solution:

(d) 66 × 22

Solution:

Question 4.

Solve the following problems:

(a) A movie theatre has 8 rows of seats, and each row has 12 seats. If half the seats are filled, how many people are watching the movie? If 3 more rows get filled, how many total people will be there?

Solution:

The movie theatre has 8 rows of seats, with 12 seats in each row.
Since only half the seats are occupied, 4 out of the 8 rows are filled.

The number of people who watched the movie is calculated by multiplying the number of filled rows by the number of seats in each row:
4 × 12 = 48 people.

If 3 additional rows are filled, the total number of occupied rows becomes 7.
So, the number of people watching the movie then is:
7 × 12 = 84 people.

(b) In a test match between India and West Indies, the Indian team hit twenty- four 4s and eighteen 6s across the two innings. How many runs were scored in 4s and 6s each? 234 runs were made by running between the wickets. If 23 runs were extras, how many runs were scored by Indian team in the two innings?

Solution:

In the test match, the Indian team hit 24 fours, which resulted in
24 × 4 = 96 runs.

They also hit 18 sixes during the match.

The Indian team scored 18 sixes, giving
18 × 6 = 108 runs.

In addition, 234 runs were made by running between the wickets, and 23 runs came from extras.

Hence, the total runs scored by the Indian team in the two innings are:
96 + 108 + 234 + 23 = 461 runs.

(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?

Solution:

The cost of 15 bulbs at ₹25 each is:

15 × ₹25 = ₹375.

The total cost is calculated as:

12 × ₹34 = ₹408.

The combined cost of 15 bulbs and 12 tube lights is:
₹375 + ₹408 = ₹783.

Therefore, Anjali needs to pay ₹783 to the shopkeeper for purchasing 15 bulbs and 12 tube lights.

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

Solution:

Each bag of rice costs ₹350.
If a shopkeeper sells 28 bags, the total amount earned is calculated as:

350 × 28 = ₹9800

Hence, by selling 28 bags of rice, the shopkeeper earns ₹9800.

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

the library.

Solution:

The library has 86 shelves, and each shelf contains 162 books.
To find the total number of books, multiply the number of shelves by the number of books per shelf:

162 × 86 = 13,932

Therefore, the library has a total of 13,932 books.

NCERT Textbook Pages 83-84

Let Us Do

Question 1.
Solve the following problems like Nida did.

a) 548 × 6

Solution:

b) 682 × 3

Solution:

c) 324 × 18

Solution:

d) 507 × 23

Solution:

e) 190 × 65

Solution:

Question 2.

Solve the following problems like John.

a) 123 × 84

Solution:

b) 368 × 32

Solution:

c) 159 × 324

Solution:

d) 239 × 401

Solution:

e) 592 × 5

Solution:

f) 101 × 22

Solution:

Question 3.

Let us solve a few questions like Mill’s father.

Solution:

Now use Mili’s father’s method to solve the following questions.

(a) 807 × 5

Solution:

(b) 143 × 28

Solution:

(c) 309 × 9

Solution:

(d) 450 × 38

Solution:

(e) 584 × 23

Solution:

(f) 302 × 13

Solution:

(g) 604 × 54

Solution:

(h) 112 × 23

Solution:

(i) 237 × 19

Solution:

NCERT Textbook Page 85

Check, Check!

Check if the following children’s solutions are correct. If correct, explain why the solution is correct. If it is incorrect, then identify the error and correct the solution.

(a) Asma’s solutions for 46 × 59

Solution:

(b) Pankaj’s solution for 203 × 54

Solution:

In Pankaj’s method for calculating 203 × 54, he makes an error in forming the partial products.
Instead of using 200, he uses 20, and instead of 50, he uses 5 at each step.
Because of this mistake, the final product obtained by him is incorrect.

Correct method:

c) Lado’s solution for 38 × 150

Solution:

She most likely broke 38 into 30 + 8 and 150 into 100 + 50.
Using this method, her calculation is correct.

d) Kira’s solution for 193 × 272

Solution:

Kira’s solution:

Kira split the first number correctly. However, while working with the second number, she used 7 instead of 70 in the partial product.

Because of this error, her final answer is incorrect.

e) Asher’s solution

Solution:

Asher made a mistake in identifying the place values of 3 and 2 in the number 323.
After correcting the place values, the correct calculation is:

626 × 323 = 2,02,198

NCERT Textbook Pages 86-87

Let Us Do

Question 1.
Identify the problems that have the same answer as the one given at the top of each box. Do not calculate.

Solution:

Question 2.

Find easy ways of solving these problems.

(a) 16 × 25

Solution:

(b) 12 × 125

Solution:

(c) 24 × 250

Solution:

(d) 36 × 25

Solution:

(e) 28 × 75

Solution:
28 × 75 = 30 × 75 – 150
= 2250 – 150
= 2100

(f) 300 × 15

Solution:
300 × 15 = 4500

(g) 50 × 78

Solution:

(h) 199 × 63

Solution:
199 × 63 = 200 × 63 – 63
= 12600 – 63
= 12537

(i) 128 × 35

Solution:
128 × 35 = 130 × 35 – 70
= 4550 – 70
= 4480

Question 3.
Write 5 other examples for which you can find easy ways of getting products.

Solution:
(a) 201 × 19
= 200 × 19 + 19
= 3800 + 19
= 3819

(b) 149 × 25
= 150 × 25 – 25
= 3750 – 25
= 3725

(c) 25 × 78

d) 28 × 18
= 30 × 18 – 18 × 2
= 540 – 36
= 504

(e) 998 × 4
= 1000 × 4 – 8
= 4000 – 8
= 3992
(Answer might be different)

Question 4.

Find the answers to the following questions based on the given information.

(a) 17 × 23 = 391

(b) 17 × 24 = _______

(c) 17 × 22 = _______

(d) 16 × 23 = _______

(e) 8 × 9 = 72

(f) 18 × 9 = _______

(g) 28 × 9 = _______

(h) 108 × 9 = _______

(i) 18 × 23 = _______

Solution:

(a) 17 × 23 = 391
(b) 17 × 24 = 408
(c) 17 × 22 = 374
(d) 16 × 23 = 368
(e) 8 × 9 = 72
(f) 18 × 9 = 162
(g) 28 × 9 = 252
(h) 108 × 9 = 972

To find 17 × 24, how much is to be added to 17 × 23 _____ 17 or 23?

Solution:

17 × 24 = 17 × 23 + (17) = 391 + (17)

To find 18 × 23, how much is to be added to 17 × 23 _____ 17 or 23?

Solution:
18 × 23 = 17 × 23 + (23)
= 391 + (23)
= 414

NCERT Textbook Pages 87-88

Let Us Think

Question 1.
Find the possible values of the coloured boxes in each of the following problems. The same colour indicates the same number in a problem. Some problems can have more than one answer.

Solution:

Question 2.

Estimate the products on the left and match them to the numbers given on the right.

Solution:

NCERT Textbook Pages 88-89

The King’s Reward

One day, a king decided to reward three of his most talented ministers. The king called them to his court and said, “You all have served my empire with great dedication. As a reward, I give you three choices of gold.

Which of the rewards would you have chosen?

Solution:
Students should do it by themselves.

After a week, the 3 ministers were surprised at the final amount of gold coins. Guess who received the most gold coins? Calculate how much gold

Solution:

Minister-1
5 × 1 = 5 → Day 1
5 × 2 = 10 → Day 2
10 × 2 = 20 → Day 3
20 × 2 = 40 → Day 4
40 × 2 = 80 → Day 5
80 × 2 = 160 → Day 6
160 × 2 = 320 → Day 7

Minister-2
3 × 1 = 3 → Day 1
3 × 3 = 9 → Day 2
9 × 3 = 27 → Day 3
27 × 3 = 851 → Day 4
81 × 3 = 243 → Day 5
243 × 3 = 729 → Day 6
729 × 3 = 2187 → Day 7

Minister-3
1 × 1 = 1 → Day 1
1 × 5 = 5 → Day 2
5 × 5 = 25 → Day 3
25 × 5 = 125 → Day 4
125 × 5 = 625 → Day 5
625 × 5 = 3125 → Day 6
3125 × 5 = 15625 → Day 7

Minister 1 received 320 gold coins, Minister 2 received 2187 gold coins, and Minister 3 received 15,625 gold coins.

Therefore, Minister 3 received the highest number of gold coins.

NCERT Textbook Pages 89-90

Multiplication Patterns

Question 1.
Notice how the multiplier, multiplicand, and products are changing in each of the following. What is the relationship of the new product with the original product? - Solve (a) completely, and then predict the answers for the rest.
a) 16 × 44 = 704
1) 8 × 88 = 704
2) 8 × 22 = 176
3) 16 × 22 = ______
4) 32 × 44 = ______

Solution:
16 × 44 = 704 .
(1) 8 × 88 = 704
(2) 8 × 22 = 176
(3) 16 × 22 = 352
(4) 32 × 44 = 1408

b) 12 × 32 = 384
1) 6 × 16 = ______
2) 24 × 16 = ______
3) 24 × 64 = ______
4) 12 × 16 = ______

Solution:
12 × 32 = 384
(1) 6 × 16 = 96
(2) 24 × 16 = 384
(3) 24 × 64 = 1536
(4) 12 × 16 = 192

Question 2.

Observe and complete the given patterns.

Solution:

Here are some numbers.

Remember number pairs from Grade 4? Any two adjacent numbers in a row or a column are number pairs. Can you identify the pair whose product is the smallest and another pair whose product is the largest? Do you need to find every product or can you find this by looking at the numbers?

Solution:
Students should do it by themselves.

NCERT Textbook Page 91

Let Us Solve

Question 1.
Mala went to a book exhibition and bought 18 books. The shop was selling 3 books for ₹ 150. After buying the books, she still had ₹ 20 left. How much money did Mala have at the beginning?

Solution:

Mala bought 18 books.
The cost of 3 books is ₹150, so the price of 1 book is ₹50.

The total cost of 18 books is calculated as:
18 × 50 = (20 × 50) − 100
= 1000 − 100
= ₹900

Since Mala still had ₹20 remaining after the purchase,
the amount she originally had was ₹920 (₹900 + ₹20).

Question 2.

A village sports club organises a women’s football tournament. The club earned money by selling match tickets and charging fees for team participation.

They sold 57 tickets for ₹ 115 each.

They had 3 teams joining the tournament, with each team paying a participation fee of ₹ 1,599.

The teams paid ₹ 1,750 in total rent the football ground and ₹ 1,129 for food and water.

(a) How much money did the club collect in total from ticket sales and team participation fees?

(b) What were the total expenses on renting the ground and food and water?

Solution:

The money collected from ticket sales is calculated as follows:
₹(57 × 115) = ₹(60 × 115 − 3 × 115)
= ₹(6900 − 345)
= ₹6555.

The total participation fee paid by the 3 teams is:
₹(3 × 1599) = ₹(3 × 1600 − 3)
= ₹4797.

The total rent paid for the football ground by the teams is ₹1750, and the cost of food and water for the teams is ₹1129.

(a) The total amount collected by the club from ticket sales and team participation fees is:
₹6555 + ₹4797 = ₹11,352.

(b) The total expenditure on ground rent and food and water is:
₹1750 + ₹1129 = ₹2879.

Question 3.

Ananya is watching Republic Day celebrations on city’s public ground. There are 12 rows of students sitting in front of her and 17 rows behind her. There are 18 students to her right and 22 students to her left.

(a) How many rows of students are there in total?

(b) How many students are there in Ananya’s row?

(c) What is the total number of students on the ground?

Solution:

(a)
The total number of rows of students on the ground is:
12 + 1 + 17 = 30 rows.

(b)
The number of students in Ananya’s row is:
18 + 1 + 22 = 41 students.

(c)
The total number of students present on the ground is:
30 × 41 = 1230 students

Question 4.

Multiply.

(a) 67 × 78

(b) 34 × 56

(c) 45 × 263

(d) 86 × 542

(e) 432 × 107

(f) 310 × 120

Solution:

(a) 67 × 78 = 67 × 80 – 2 × 67
= 5360 – 134
= 5226

(b) 34 × 56 = 30 × 56 + 4 × 56
= 1680 + 224
= 1904

(c) 45 × 263 = 40 × 263 + 5 × 263
= 10,520 + 1315
= 11,835

(d) 86 × 542 = 86 × (540 + 2)
= 86 × 540 + 2 × 86
= 46,440 + 172
= 46,612

(e) 432 × 107 = 432 × 100 + 432 × 7
= 43,200 + 3024
= 46,224

(f) 310 × 120 = 300 × 120 + 10 × 120
= 36,000 + 1200
= 37,200

Question 5.
If 67 × 67 = 4489, without multiplication find 67 × 68.

Solution:
Since, 67 × 67 = 4489
∴ 67 × 68
= 67 × 67 + 67
= 4489 + 67 = 4556

Question 6.

If 99 × 100 = 9900, without multiplication find 99 × 99.

Solution:

Since, 99 × 100 = 9900

∴ 99 × 99

= 99 × 100 – 99

= 9900 – 99

= 9801

Mastering Multiplication in The Dairy Farm – Class 5 Maths

Learning multiplication strategies in NCERT Solutions Class 5 Maths Mela Chapter 6 The Dairy Farm makes calculations faster and easier. By practicing regularly, you build a strong foundation in basic arithmetic—a skill that will help in all higher maths topics.

This chapter focuses on using patterns, doubling, halving, and nearest multiples to solve problems efficiently. Understanding such techniques is essential for students aiming for high exam scores and for building conceptual clarity.

Practice every section of NCERT Solutions Class 5 Maths Mela Chapter 6 The Dairy Farm 2025-26 for confidence in exams. Revising worked examples and creating your own helps you tackle both direct and story-based questions easily.