Class 5 Mathematics · Chapter 6

6. The Dairy Farm

Solutions to all Let Us Think, Let Us Solve, Let Us Do and Check, Check! questions. CBSE / NCERT Class 5 Maths Mela.

Let Us Think

1The 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.

From (d): diamond × pentagon = diamond. So, pentagon = 1.

From (b): rectangle × rectangle is at most 24. So, the rectangle is 2, 3 or 4.

One answer that fits all the problems:

Shape	Number
Rectangle	3
Circle	4
Triangle	12
Badge shape	9
Parallelogram	2
Diamond	6
Pentagon	1
Half circle	8

Check: (a) 3 × 4 = 12 (b) 3 × 3 = 9 (c) 2 × 6 = 12 (d) 6 × 1 = 6 (e) 2 × 4 = 8 (f) 8 × 3 = 2 × 12 = 24

Another answer: rectangle 4, circle 3, triangle 12, badge 16, parallelogram 6, diamond 2, pentagon 1, half circle 18.

2Place the digits 2, 5 and 3 appropriately to get a product close to 100.

53 × 2 = 106

Other products are 25 × 3 = 75, 35 × 2 = 70, 23 × 5 = 115, 52 × 3 = 156 and 32 × 5 = 160. So, 106 is the closest to 100.

3A dairy has packed butter milk pouches in different ways. Find the number of pouches kept in each arrangement.

30 × 2 = 60 (given)

20 × 3 = 60 (20 groups of 3)

6 × 10 = 60 (6 groups of 10)

5 × 12 = 60 (5 rows of 12)

Other groups: 4 × 15, 15 × 4, 3 × 20, 1 × 60 and 60 × 1.

4Which number am I? I am a two-digit number. Find me with the help of the clues.

Multiples of 9 that are not multiples of 4: 18, 27, 45, 54, 63, 81, 90, 99.

Less than 50 and not a multiple of 11: 18, 27, 45.

Ones digit even and tens digit odd: 18.

The number is 18.

Clue (d) says “I am an odd number”, but an odd number cannot have an even ones digit (clue g). So clue (d) seems to be a printing error — 18 is even and fits all the other clues. Clues (a), (e) and (h) did not help to find the number.

5Make your own numbers. Choose any two numbers and one operation from the grid to make the numbers 0 to 20.

Number	How to make it
1	5 − 4, 4 − 3, 3 − 2
2	4 − 2, 5 − 3, 10 ÷ 5, 12 − 10
3	5 − 2, 12 ÷ 4, 36 ÷ 12
4	12 ÷ 3, 100 ÷ 25
5	2 + 3, 10 − 5, 25 ÷ 5, 10 ÷ 2
6	2 × 3, 2 + 4, 10 − 4, 12 ÷ 2
7	5 + 2, 4 + 3, 10 − 3, 12 − 5
8	5 + 3, 2 × 4, 10 − 2, 12 − 4
9	5 + 4, 12 − 3, 36 ÷ 4
10	5 × 2, 12 − 2, 100 ÷ 10
11	36 − 25
12	10 + 2, 4 × 3, 36 ÷ 3
13	10 + 3, 25 − 12
14	10 + 4, 12 + 2
15	5 × 3, 10 + 5, 12 + 3, 25 − 10
16	12 + 4
17	12 + 5
18	36 ÷ 2
20	5 × 4, 10 × 2, 25 − 5, 100 ÷ 5

Numbers we could not make: 0 and 19.

Using three numbers: 0 = 5 − 2 − 3 and 19 = 10 + 5 + 4 (or 25 − 4 − 2).

Numbers we can get in more than one way: 1 to 10, 12, 13, 14, 15 and 20.

Order of Numbers in Multiplication

1Find the number of butter packets in each case. What pattern do you notice?

a) 2 × 3 = 6, 3 × 2 = 6
b) 8 × 5 = 40, 5 × 8 = 40

6 × 13 = 78, 13 × 6 = 78

e) 10 × 5 = 50, 5 × 10 = 50
f) 8 × 20 = 160, 20 × 8 = 160
g) 12 × 9 = 108, 9 × 12 = 108

9 × 0 = 0 and 0 × 9 = 0.

Pattern: Changing the order of the numbers does not change the product. Yes, this is true for any two numbers.

Patterns in Multiplication by 10s and 100s

1Let us revise multiplication by 10s and 100s.

a) 4 × 10 = 40
b) 20 × 10 = 200
c) 10 × 40 = 400

e) 20 × 50 = 1,000
f) 80 × 10 = 800

h) 8 × 100 = 100 × 8 = 800
i) 10 × 100 = 100 × 10 = 1,000

2Find the answers, fill in the table and describe the pattern.

100 × 90 = 9,000
400 × 10 = 4,000
60 × 50 = 3,000
700 × 4 = 2,800
10 × 45 = 450

Problem	Th	H	T
10 × 45 = 450		4	5
30 × 20 = 600		6	0
400 × 10 = 4,000	4	0	0
700 × 8 = 5,600	5	6	0
100 × 90 = 9,000	9	0	0
60 × 50 = 3,000	3	0	0
220 × 20 = 4,400	4	4	0
11 × 300 = 3,300	3	3	0
80 × 90 = 7,200	7	2	0
10 × 63 = 630		6	3
40 × 12 = 480		4	8

Pattern: Multiply the non-zero digits first. Then write as many zeros as there are in both numbers together. For example, 60 × 50: 6 × 5 = 30, and two zeros → 3,000.

3Multiplying by 1,000s — fill in the table and observe the patterns.

Problem	Product
3 × 5,000	15,000
8 × 3,000	24,000
5 × 7,000	35,000
20 × 100	2,000
40 × 500	20,000
60 × 300	18,000
600 × 30	18,000
80 × 900	72,000
70 × 600	42,000

Many Ways to Multiply

1What is 18 × 5? Do you think all the children’s methods are correct? Why?

18 × 5 = 90.

Yes, all the methods are correct. Each child broke the numbers in a different way, but the total is the same.

Doubling and Halving

1Find strategies to calculate the total number of butter packets.

a) 3 × 18 = 6 × 9 = 54. They are the same because when one number is doubled and the other is halved, the product does not change.

b) 22 × 5 = 11 × 10 = 110

2Solve the following problems like the previous ones.

14 × 3 = 7 × 6 = 42
38 × 5 = 19 × 10 = 190

16 × 4 = 8 × 8 = 64
35 × 14 = 70 × 7 = 490

3This halving and doubling strategy works well with numbers like 5 and 25. Discuss why.

Double of 5 is 10, and 4 times 25 is 100. Multiplying by 10 or 100 is very easy.

4(d) Find the product by halving and doubling either the multiplier or the multiplicand.

1) 5 × 18 = 10 × 9 = 90
2) 50 × 28 = 100 × 14 = 1,400
3) 15 × 22 = 30 × 11 = 330

4) 25 × 12 = 50 × 6 = 100 × 3 = 300
5) 12 × 45 = 6 × 90 = 540
6) 16 × 45 = 8 × 90 = 720

5(e) Give 5 examples where halving and doubling help. Find the products.

5 × 44 = 10 × 22 = 220

25 × 16 = 50 × 8 = 100 × 4 = 400

50 × 36 = 100 × 18 = 1,800

15 × 18 = 30 × 9 = 270

35 × 8 = 70 × 4 = 280

Nearest Multiple

1Why do we subtract 4 in 4 × 19 and add 14 in 14 × 21?

4 × 19 = 4 × 20 − 4. In 4 × 20 we took one extra column of 4. So, we subtract 4. 80 − 4 = 76.

14 × 21 = 14 × 20 + 14. 14 × 20 has one column less. So, we add 14. 280 + 14 = 294.

2(c) Give 5 examples where you can use the nearest multiple. Find the products.

6 × 49 = 6 × 50 − 6 = 294

8 × 31 = 8 × 30 + 8 = 248

9 × 99 = 9 × 100 − 9 = 891

12 × 11 = 12 × 10 + 12 = 132

25 × 19 = 25 × 20 − 25 = 475

3(d) Find the products by finding the nearest multiple.

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

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

3) 75 × 31 = 75 × 30 + 75 = 2,250 + 75 = 2,325

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

5) 8 × 25 = 8 × 24 + 8 = 192 + 8 = 200

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

Let Us Solve

1An auditorium has 35 rows with 42 seats in each row. How many people can sit in it?

35 × 42 = 35 × 40 + 35 × 2 = 1,400 + 70 = 1,470 people

2Priya jogs 4 km every day. How many km will she jog in 31 days?

4 × 31 = 4 × 30 + 4 = 124 km

3A school received 36 boxes with 48 books in each box. How many books did it receive?

36 × 48 = 36 × 50 − 36 × 2 = 1,800 − 72 = 1,728 books

4Priya uses 16 m of cloth to make 4 kurtas. How much cloth for 8 kurtas?

8 kurtas = 2 × 4 kurtas. Cloth needed = 2 × 16 m = 32 m

5Gollappa has 29 cows. Each cow gives 5 litres of milk per day. How much milk in total each day?

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

6Maska Cow Farm has 297 cows. Each cow needs 18 kg of fodder per day. How much fodder is needed every day?

297 × 18 = 300 × 18 − 3 × 18 = 5,400 − 54 = 5,346 kg

Waste and Composting

1A family produces 35 kg of kitchen waste in a month. How much in a year? How are Nida’s, Kanti’s and John’s solutions the same or different?

12 × 35 = 300 + 50 + 60 + 10 = 420. The family produces 420 kg of waste in a year.

All three break 35 into 30 + 5 and 12 into 10 + 2. Nida uses a grid. Kanti writes all four parts separately. John adds the parts row by row (2 × 35 = 70 and 10 × 35 = 350). All get 420.

2The family gets 150 kg of compost each year. 1 kg is sold at ₹24. How much money do they save?

₹24 × 150 = ₹3,600. The family saves ₹3,600.

Let Us Multiply

(a)32 × 8

Kanti’s way: 16 (8 × 2) + 240 (8 × 30) = 256

John’s way: 240 + 16 = 256

(b)69 × 45

Grid: 40 × 60 = 2,400, 40 × 9 = 360, 5 × 60 = 300, 5 × 9 = 45

Kanti’s way: 45 + 300 + 360 + 2,400 = 3,105

John’s way: 300 + 45 = 345 (5 × 69); 2,400 + 360 = 2,760 (40 × 69); 345 + 2,760 = 3,105

Let Us Do

1Solve the following problems like Nida did.

a) 78 × 4

×	70	8
4	280	32

280 + 32 = 312

b) 83 × 9

×	80	3
9	720	27

720 + 27 = 747

c) 67 × 28

×	60	7
20	1,200	140
8	480	56

1,340 + 536 = 1,876

d) 53 × 37

×	50	3
30	1,500	90
7	350	21

1,590 + 371 = 1,961

2Solve the following problems like Kanti.

a) 94 × 5 = 20 (5 × 4) + 450 (5 × 90) = 470

b) 49 × 6 = 54 (6 × 9) + 240 (6 × 40) = 294

c) 37 × 53 = 21 (3 × 7) + 90 (3 × 30) + 350 (50 × 7) + 1,500 (50 × 30) = 1,961

d) 28 × 79 = 72 (9 × 8) + 180 (9 × 20) + 560 (70 × 8) + 1,400 (70 × 20) = 2,212

3Solve the following problems like John.

a) 86 × 3 = 240 + 18 = 258

b) 72 × 7 = 490 + 14 = 504

c) 94 × 36: 540 + 24 = 564 (6 × 94); 2,700 + 120 = 2,820 (30 × 94); total = 3,384

d) 66 × 22: 120 + 12 = 132 (2 × 66); 1,200 + 120 = 1,320 (20 × 66); total = 1,452

4Solve the following problems.

(a) A theatre has 8 rows with 12 seats each. If half the seats are filled, how many people are watching? If 3 more rows get filled, how many in total?

Total seats = 8 × 12 = 96. Half = 96 ÷ 2 = 48 people.

3 more rows = 3 × 12 = 36. Total = 48 + 36 = 84 people.

(b) India hit twenty-four 4s and eighteen 6s. 234 runs were made by running and 23 were extras. Find the runs in 4s, in 6s, and the total.

Runs in 4s = 24 × 4 = 96. Runs in 6s = 18 × 6 = 108.

Total runs = 96 + 108 + 234 + 23 = 461 runs

15 × ₹25 = ₹375. 12 × ₹34 = ₹408. Total = ₹783

(d) A shopkeeper sold 28 bags of rice at ₹350 each. How much did he earn?

28 × ₹350 = ₹9,800

(e) A library has 86 shelves with 162 books each. Find the number of books.

86 × 162 = 12,960 + 972 = 13,932 books

Dairy Cooperative

1268 villagers each have at least 4 cows. 453 Gir cows give 13 litres each. Ghee sells at ₹574 per kg for 125 kg. Find the answers.

Minimum cows = 268 × 4 = 800 + 240 + 32 = 1,072 cows

Milk from Gir cows = 453 × 13 = 1,359 + 4,530 = 5,889 litres

Money from ghee = ₹574 × 125 = 2,870 + 11,480 + 57,400 = ₹71,750

Let Us Solve

1Solve the following problems like Nida.

a) 548 × 6 = 3,000 + 240 + 48 = 3,288

b) 682 × 3 = 1,800 + 240 + 6 = 2,046

c) 324 × 18 = (3,000 + 200 + 40) + (2,400 + 160 + 32) = 3,240 + 2,592 = 5,832

d) 507 × 23 = (10,000 + 0 + 140) + (1,500 + 0 + 21) = 10,140 + 1,521 = 11,661

e) 190 × 65 = (6,000 + 5,400) + (500 + 450) = 11,400 + 950 = 12,350

2Solve the following problems like John.

a) 123 × 84 = 492 (4 × 123) + 9,840 (80 × 123) = 10,332

b) 368 × 32 = 736 (2 × 368) + 11,040 (30 × 368) = 11,776

c) 159 × 324 = 636 (4 × 159) + 3,180 (20 × 159) + 47,700 (300 × 159) = 51,516

d) 239 × 401 = 239 (1 × 239) + 95,600 (400 × 239) = 95,839

e) 592 × 5 = 2,500 + 450 + 10 = 2,960

f) 101 × 22 = 202 (2 × 101) + 2,020 (20 × 101) = 2,222

3Solve like Mili’s father.

806 × 9 = 7,254
455 × 26 = 2,730 + 9,100 = 11,830
143 × 208 = 1,144 + 28,600 = 29,744

(a) 807 × 5 = 4,035
(b) 143 × 28 = 1,144 + 2,860 = 4,004
(c) 309 × 9 = 2,781

(d) 450 × 38 = 3,600 + 13,500 = 17,100
(e) 584 × 23 = 1,752 + 11,680 = 13,432
(f) 302 × 13 = 906 + 3,020 = 3,926

(g) 604 × 54 = 2,416 + 30,200 = 32,616
(h) 112 × 23 = 336 + 2,240 = 2,576
(i) 237 × 19 = 2,133 + 2,370 = 4,503

Check, Check!

a)Asma’s solution for 46 × 59 = 2,714

Correct. 40 × 50 = 2,000, 6 × 50 = 300, 40 × 9 = 360, 6 × 9 = 54. 2,000 + 300 + 360 + 54 = 2,714.

b)Pankaj’s solution for 203 × 54 = 135

Incorrect. He forgot the place value of the digits.

4 × 203 = 800 + 12 = 812 (not 80 + 12). 50 × 203 = 10,000 + 150 = 10,150 (not 100 + 15).

Correct answer: 812 + 10,150 = 10,962

c)Lado’s solution for 38 × 150 = 5,700

Correct. 50 × 38 = 1,500 + 400 = 1,900. 100 × 38 = 3,000 + 800 = 3,800. 1,900 + 3,800 = 5,700.

d)Kira’s solution for 193 × 272 = 40,337

Incorrect. The second row should be 70 × 193, not 7 × 193.

70 × 193 = 7,000 + 6,300 + 210 = 13,510.

Correct answer: 386 + 13,510 + 38,600 = 52,496

e)Asher’s solution for 626 × 323 = 5,018

Incorrect. He did not shift the rows for tens and hundreds.

3 × 626 = 1,878. 20 × 626 = 12,520. 300 × 626 = 1,87,800.

Correct answer: 1,878 + 12,520 + 1,87,800 = 2,02,198

Let Us Do

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

12 × 17 → 6 × 34 (half and double)

26 × 11 → 26 × 10 and 26 × 1 and 20 × 11 and 6 × 11 (both)

18 × 4 → 9 × 8 and 20 × 4 − 8 (both)

55 × 9 → 50 × 9 and 5 × 9 and 55 × 10 − 55

101 × 42 → 100 × 42 and 42

247 × 8 → 250 × 8 − 24

1001 × 5 → 1,000 × 5 and 5

1999 × 2 → 2,000 × 2 − 2

2Find easy ways of solving these problems.

(a) 16 × 25 = 4 × 100 = 400
(b) 12 × 125 = 3 × 500 = 1,500
(c) 24 × 250 = 6 × 1,000 = 6,000

(d) 36 × 25 = 9 × 100 = 900
(e) 28 × 75 = 7 × 300 = 2,100
(f) 300 × 15 = 45 × 100 = 4,500

(g) 50 × 78 = 100 × 39 = 3,900
(h) 199 × 63 = 200 × 63 − 63 = 12,537
(i) 128 × 35 = 64 × 70 = 4,480

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

25 × 44 = 100 × 11 = 1,100; 99 × 7 = 700 − 7 = 693; 15 × 16 = 30 × 8 = 240; 125 × 8 = 1,000; 49 × 5 = 50 × 5 − 5 = 245

4Find the answers based on the given information.

(b) 17 × 24 = 391 + 17 = 408
(c) 17 × 22 = 391 − 17 = 374
(d) 16 × 23 = 391 − 23 = 368

(f) 18 × 9 = 72 + 90 = 162
(g) 28 × 9 = 72 + 180 = 252
(h) 108 × 9 = 72 + 900 = 972

(i) 18 × 23 = 391 + 23 = 414

To find 17 × 24, add 17 to 17 × 23. To find 18 × 23, add 23 to 17 × 23.

Let Us Think

1Find the possible values of the coloured boxes. The same colour indicates the same number.

a) 11 × 3 = 33, 22 × 3 = 66, 33 × 3 = 99

b) 111 × 3 = 333, 222 × 3 = 666, 333 × 3 = 999

c) 1,111 × 3 = 3,333, 2,222 × 3 = 6,666, 3,333 × 3 = 9,999

d) 26 × 5 = 130, 28 × 5 = 140, 30 × 5 = 150, 32 × 5 = 160, 34 × 5 = 170, 36 × 5 = 180, 38 × 5 = 190

e) 110 × 5 = 550

f) 3,332 × 5 = 16,660

g) 202 × 9 = 1,818, 303 × 9 = 2,727, 404 × 9 = 3,636, 505 × 9 = 4,545 (and so on up to 909 × 9 = 8,181)

h) 857 × 7 = 5,999

i) No 3-digit number fits all the colours.

In (i), 2,000 can only be made as 1,000 × 2, 500 × 4, 400 × 5 or 250 × 8. In none of these is the middle digit the same as the multiplier, so this puzzle seems to have a printing error.

2Estimate the products and match them to the numbers.

25 × 31 ≈ 25 × 30 → 750

132 × 19 ≈ 130 × 20 → 2,600

101 × 11 ≈ 100 × 10 → 1,000

248 × 49 ≈ 250 × 50 → 12,500

12 × 25 → 300

The King’s Reward

1Which reward would you have chosen? Who received the most gold coins? Calculate the coins each minister received.

Day	Minister 1 (× 2)	Minister 2 (× 3)	Minister 3 (× 5)
Start	5	3	1
Day 1	10	9	5
Day 2	20	27	25
Day 3	40	81	125
Day 4	80	243	625
Day 5	160	729	3,125
Day 6	320	2,187	15,625
Day 7	640	6,561	78,125

Minister 1 gets 640 coins, Minister 2 gets 6,561 coins and Minister 3 gets 78,125 coins.

Minister 3 received the most gold coins. I would choose Choice 3.

Multiplication Patterns

1What is the relationship of the new product with the original product?

a) 16 × 44 = 704

1) 8 × 88 = 704 — one number halved, the other doubled. Product stays the same.

2) 8 × 22 = 176 — both halved. Product becomes one-fourth.

3) 16 × 22 = 352 — one number halved. Product becomes half.

4) 32 × 44 = 1,408 — one number doubled. Product becomes double.

b) 12 × 32 = 384

1) 6 × 16 = 96 (one-fourth)
2) 24 × 16 = 384 (same)

3) 24 × 64 = 1,536 (4 times)
4) 12 × 16 = 192 (half)

2Observe and complete the given patterns.

1111 × 1111 = 12,34,321

45 × 45 = 2,025 (4 × 5 = 20, then 25)
55 × 55 = 3,025 (5 × 6 = 30, then 25)

11 × 54 = 594 (5, 5 + 4, 4)
11 × 82 = 902 (8, 8 + 2 = 10, 2 → 902)

1,234 × 9 + 4 = 11,110

3Identify the number pair whose product is the smallest and the pair whose product is the largest.

Smallest product: 11 and 14 → 11 × 14 = 154. Both are small numbers.

Largest product: 35 and 75 → 35 × 75 = 2,625.

We do not need to find every product. We look for the two smallest and the two biggest numbers that are next to each other.

Let Us Solve

1Mala bought 18 books. The shop sold 3 books for ₹150. She still had ₹20 left. How much money did she have at the beginning?

18 books = 6 sets of 3 books. Cost = 6 × ₹150 = ₹900

Money at the beginning = ₹900 + ₹20 = ₹920

2A village sports club organises a women’s football tournament.

(a) Tickets: 57 × ₹115 = ₹6,555. Team fees: 3 × ₹1,599 = ₹4,797. Total collected = ₹11,352

(b) Total expenses = ₹1,750 + ₹1,129 = ₹2,879

3Ananya has 12 rows in front, 17 rows behind, 18 students to her right and 22 to her left.

(a) Total rows = 12 + 1 + 17 = 30 rows

(b) Students in her row = 18 + 1 + 22 = 41 students

4Multiply.

(a) 67 × 78 = 5,226
(b) 34 × 56 = 1,904
(c) 45 × 263 = 11,835

(d) 86 × 542 = 46,612
(e) 432 × 107 = 46,224
(f) 310 × 120 = 37,200

5If 67 × 67 = 4,489, without multiplication find 67 × 68.

67 × 68 = 67 × 67 + 67 = 4,489 + 67 = 4,556

6If 99 × 100 = 9,900, without multiplication find 99 × 99.

99 × 99 = 99 × 100 − 99 = 9,900 − 99 = 9,801

← 5. Far and Near 7. Shapes and Patterns →