NCERT Class 5 Maths · Chapter 6

The Dairy Farm

Many smart ways to multiply: order, tens and hundreds, doubling and halving, nearest multiples, the box method, and column multiplication.

📌 What you’ll learn in this chapter
  • Changing the order doesn’t change the product (6 × 13 = 13 × 6)
  • Multiplying by 10, 100, and 1,000 quickly
  • Shortcuts: doubling and halving, and using the nearest multiple
  • Multiplying big numbers by breaking them into parts (grid/box method)
  • The standard column method, and checking answers

🐄 Let’s Begin: Counting at the Dairy

Kamla Didi runs a small dairy in her village. Every morning, 24 families bring milk. Each family brings 15 litres. “How much milk do we collect every day?” she wonders.

She could add 15 twenty-four times… or she could multiply: 24 × 15. Let’s learn many clever ways to do it!

🔄 Part 1: Order Doesn’t Matter

Arrange 20 ladoos in 4 rows of 5 (4 × 5). Now turn the tray sideways: it’s 5 rows of 4 (5 × 4). Still 20 ladoos!

4 × 5 = 5 × 4 = 20

So you can always choose the easier order. 7 × 100 is hard to “count 7 hundred times”, but 100 × 7 = 700 is easy.

💡 Zero and one Any number × 0 = 0 (9 × 0 = 0 × 9 = 0). Any number × 1 stays the same.

🔟 Part 2: Multiplying by 10s, 100s, and 1,000s

  • × 10 → each digit moves one place left (add one zero): 34 × 10 = 340
  • × 100 → two places left: 34 × 100 = 3,400
  • × 1,000 → three places left: 34 × 1,000 = 34,000

For numbers like 40 × 70

Multiply the non-zero parts, then count the zeros: 4 × 7 = 28, and there are 2 zeros → 2,800.

More: 300 × 60 → 3 × 6 = 18, 3 zeros → 18,000 · 50 × 400 → 5 × 4 = 20, 3 zeros → 20,000

⚠️ Common mistake 50 × 40 = 200 ❌ (forgot a zero from 5 × 4 = 20)
✅ 5 × 4 = 20, then add 2 zeros → 2,000

🧠 Part 3: Smart Ways to Multiply

A. Doubling and halving

Halve one number and double the other. The answer stays the same!

16 × 5 → 8 × 10 = 80  ·  24 × 25 → 12 × 50 → 6 × 100 = 600  ·  14 × 15 → 7 × 30 = 210

Why does it work? Picture 16 rows of 5 laddoos. Cut the tray in half and put the halves side by side. Now it’s 8 rows of 10, the same laddoos! It works best with 5, 25, and 50, because doubling them gives 10, 50, and 100.

B. Nearest multiple (round, then fix)

6 × 49 → 6 × 50 = 300, but that’s one 6 too many → 300 − 6 = 294

8 × 61 → 8 × 60 = 480, and add one more 8 → 480 + 8 = 488

C. Breaking numbers (box / grid method)

Let’s solve Kamla Didi’s problem: 24 × 15. Break 24 = 20 + 4 and 15 = 10 + 5.

×204
1020040
510020

Add all the boxes: 200 + 40 + 100 + 20 = 360 litres of milk every day! 🥛

D. Row by row (partial products)

Multiply the whole top number by each part of the bottom number:

146 × 23
146 × 3 = 438
146 × 20 = 2,920
Total = 438 + 2,920 = 3,358

E. Standard column method

This is the same as D, written short. Multiply by the ones digit first, then by the tens digit (put a 0 in the ones place first, since you are multiplying by tens), then add.

146
×23
438
+2920
3358
⚠️ The most common mistake Forgetting the 0 in the second row. Then you add 438 + 292 = 730 ❌
✅ The 2 in 23 is really 20, so the second row is 2,920.

🔍 Part 4: Estimate First, Then Check

Before multiplying, round to get a rough answer. 198 × 31 is about 200 × 30 = 6,000. If your exact answer is 61,380 or 614, you know something went wrong. (The exact answer is 6,138.)

❌ Common Mistakes to Avoid

Missing the 0 when multiplying by tens
✅ Write 0 in the ones place of the second row.
Forgetting to add the carry
✅ Write carries small on top and cross them off after use.
Counting zeros wrong (60 × 50)
✅ 6 × 5 = 30, then 2 more zeros = 3,000.
Not estimating
✅ A quick estimate catches big errors.

✏️ Practice Time

Try these on your own first, then check your answer below each question.

Q1. 70 × 800 = ?
✅ Answer7 × 8 = 56, 3 zeros → 56,000
Q2. Use doubling and halving: 18 × 50
✅ Answer9 × 100 = 900
Q3. Use the nearest multiple: 9 × 98
✅ Answer9 × 100 − 9 × 2 = 900 − 18 = 882
Q4. Use the box method: 37 × 26
✅ Answer30 × 20 = 600, 7 × 20 = 140, 30 × 6 = 180, 7 × 6 = 42. Total = 962
Q5. A dairy packs 245 pouches every hour. How many in 16 hours?
✅ Answer245 × 6 = 1,470 and 245 × 10 = 2,450. So 1,470 + 2,450 = 3,920 pouches
Q6. Paneer costs ₹380 per kg. What do 125 kg cost?
✅ Answer380 × 100 = 38,000 · 380 × 20 = 7,600 · 380 × 5 = 1,900. Total = ₹47,500
Q7. Estimate: 412 × 19 is closest to (a) 800 (b) 8,000 (c) 80,000
✅ AnswerAbout 400 × 20 = 8,000, so (b). (Exact: 7,828)

📘 NCERT Textbook Activity Help

Hints and answers for the activities in the textbook, page by page. Try each one yourself before looking at the answer!

Page 70 · Let Us Think, Q1–Q3

Hidden shapes, digits, and pouches

Hint: Start with (d): a number × the pentagon gives back the same number, so the pentagon must be 1.

✅ Answer Q1 (one answer that works for all six): Pentagon = 1 · Rectangle = 3 · Circle = 4 · Triangle = 12 · Cloud shape = 9 · Parallelogram = 2 · Diamond = 6 · Half-moon = 8. Check (f): 8 × 3 = 24 and 2 × 12 = 24 ✅ (A few other answers also work.)
Q2: 53 × 2 = 106 is closest to 100
Q3: 60 = 30 × 2 = 20 × 3 = 15 × 4 = 12 × 5. Other groups: 10 × 6, 6 × 10, 5 × 12, 4 × 15, 3 × 20, 2 × 30, 1 × 60
Page 71 · Q4, Q5

Which number am I? Make your own numbers

✅ Answer Q4: Multiples of 9 below 50 that are not multiples of 4 or 11: 18, 27, 45. With “tens digit odd” and “ones digit even”, the answer is 18. (Clue (d), “I am odd”, does not fit 18, so it looks like a mistake in the clues.) Clue (a) “greater than 8” was not needed.
Q5: With two numbers, every number from 1 to 20 can be made except 0 and 19. For example, 11 = 36 − 25, 18 = 36 ÷ 2, 13 = 25 − 12. With three numbers: 19 = 25 − 10 + 4, and 0 = 5 − 2 − 3. Many numbers can be made in more than one way, like 6 = 2 × 3 = 10 − 4 = 12 ÷ 2.
Page 72 · Order of numbers

Butter packets

✅ Answer(a) 6, 6 (b) 40, 40 · 6 × 13 = 13 × 6 = 78 · (e) 50 (f) 160 (g) 108 · 9 × 0 = 0 × 9 = 0. Yes, this is true for any two numbers.
Q1: (a) 40 (b) 200 (c) 400 (e) 1,000 (f) 800 (h) 800 = 100 × 8 (i) 1,000 = 100 × 10
Pages 73–74 · Tens and hundreds

Fill the place value tables

✅ Answer 100 × 90 = 9,000 · 400 × 10 = 4,000 · 60 × 50 = 3,000 · 700 × 4 = 2,800 · 10 × 45 = 450 (4 H, 5 T, 0 O) · 700 × 8 = 5,600 · 220 × 20 = 4,400 · 11 × 300 = 3,300 · 80 × 90 = 7,200 · 10 × 63 = 630 · 40 × 12 = 480
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
Page 74 · Many ways to multiply

18 × 5

✅ AnswerAll the methods are correct. Each one breaks or changes the numbers in a fair way, and every method gets 90.
Pages 75–76 · Doubling and halving

Halve one, double the other

✅ Answer (a) 3 × 18 = 6 × 9 = 54 (b) 22 × 5 = 11 × 10 = 110
(c) 14 × 3 = 7 × 6 = 42 · 38 × 5 = 19 × 10 = 190 · 16 × 4 = 8 × 8 = 64 · 35 × 14 = 70 × 7 = 490
It works well with 5 and 25 because doubling them gives 10 and 50, which are easy to multiply.
(d) 1) 90 2) 1,400 3) 330 4) 300 5) 540 6) 720
(e) Examples: 45 × 4 = 90 × 2 = 180 · 25 × 16 = 100 × 4 = 400 · 15 × 18 = 30 × 9 = 270 · 50 × 36 = 100 × 18 = 1,800 · 5 × 64 = 10 × 32 = 320
Page 76 · Nearest multiple

Round and adjust

✅ Answer (a) We multiplied 4 × 20, which counts one extra column of 4. So we subtract 4. (b) 14 × 20 misses one column of 14, so we add 14.
(c) Examples: 5 × 39 = 200 − 5 = 195 · 6 × 101 = 600 + 6 = 606 · 3 × 99 = 300 − 3 = 297 · 12 × 51 = 600 + 12 = 612 · 7 × 29 = 210 − 7 = 203
(d) 1) 364 2) 336 3) 2,325 4) 1,485 5) 200 6) 924
Page 77 · Let Us Solve

Word problems

✅ Answer1. 35 × 42 = 1,470 2. 4 × 31 = 124 km 3. 36 × 48 = 1,728 4. 8 kurtas = twice 4 kurtas = 32 m 5. 29 × 5 = 145 litres 6. 297 × 18 = 5,346 kg
Pages 77–78 · Waste and composting

Nida, Kanti, and John

✅ AnswerThe family produces 420 kg of waste in a year. All three methods break 35 into 30 + 5 and 12 into 10 + 2. Nida uses a grid, Kanti lists the 4 small products, and John multiplies all of 35 by 2 and by 10, then adds. Compost savings: ₹3,600.
Pages 79–80 · Let Us Multiply / Let Us Do

Practice with each method

✅ Answer 32 × 8: 16 (8 × 2) and 240 (8 × 30) → 256 · 69 × 45: 45, 300, 360, 2,400 → 3,105
Q1 (Nida): (a) 312 (b) 747 (c) 1,876 (d) 1,961
Q2 (Kanti): (a) 470 (b) 294 (c) 1,961 (d) 2,212
Q3 (John): (a) 258 (b) 504 (c) 3,384 (d) 1,452
Q4: (a) Half of 96 seats = 48 people. 3 more rows = 36 more, so 84 people. (b) 4s: 24 × 4 = 96 runs. 6s: 18 × 6 = 108 runs. Total = 96 + 108 + 234 + 23 = 461 runs. (c) 15 × 25 + 12 × 34 = 375 + 408 = ₹783 (d) 28 × 350 = ₹9,800 (e) 86 × 162 = 13,932 books
Pages 81–83 · Dairy cooperative

Cows, milk, and ghee

✅ Answer268 × 4 = 1,072 cows · 453 × 13 = 5,889 litres · 574 × 125 = ₹71,750. The same-coloured digits show where each carry goes in the column method.
Pages 83–84 · Let Us Solve

Bigger multiplications

✅ Answer Q1: (a) 3,288 (b) 2,046 (c) 5,832 (d) 11,661 (e) 12,350
Q2: (a) 10,332 (b) 11,776 (c) 51,516 (d) 95,839 (e) 2,960 (f) 2,222
Q3: 806 × 9 = 7,254 · 455 × 26 = 11,830 · 143 × 208 = 29,744
(a) 4,035 (b) 4,004 (c) 2,781 (d) 17,100 (e) 13,432 (f) 3,926 (g) 32,616 (h) 2,576 (i) 4,503
Page 85 · Check, Check!

Find the mistakes

✅ Answer (a) Asma: Correct. 2,000 (40 × 50) + 300 (6 × 50) + 360 (40 × 9) + 54 (6 × 9) = 2,714
(b) Pankaj: Wrong. He forgot place values. 4 × 203 = 812 (not 92), and 50 × 203 = 10,150 (not 115). Correct answer: 10,962
(c) Lado: Correct. 50 × 38 = 1,900 and 100 × 38 = 3,800, so 5,700
(d) Kira: Wrong. The middle row is for 70, not 7: 70 × 193 = 13,510 (not 1,351). Correct answer: 386 + 13,510 + 38,600 = 52,496
(e) Asher: Wrong. The second row (626 × 20) should be 12,520 and the third (626 × 300) should be 1,87,800. He did not shift the rows. Correct answer: 2,02,198
Pages 86–87 · Let Us Do

Same answers and easy ways

✅ Answer Q1: 12 × 17 → 6 × 34 · 26 × 11 → both options · 18 × 4 → both (9 × 8 and 20 × 4 − 8) · 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
Q2: (a) 400 (b) 1,500 (c) 6,000 (d) 900 (e) 2,100 (f) 4,500 (g) 3,900 (h) 12,537 (i) 4,480
Q4: (b) 391 + 17 = 408 (c) 391 − 17 = 374 (d) 391 − 23 = 368 (f) 72 + 90 = 162 (g) 162 + 90 = 252 (h) 972 (i) 391 + 23 = 414
For 17 × 24, add 17 (one more column). For 18 × 23, add 23 (one more row).
Pages 87–88 · Let Us Think

Coloured boxes and estimation

Hint: Same colour means the same digit.

✅ Answer Q1 (sample answers): (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) 24 × 5 = 120, 36 × 5 = 180 (e) 110 × 5 = 550 (f) 3,332 × 5 = 16,660 (g) 202 × 9 = 1,818, 303 × 9 = 2,727 (h) 857 × 7 = 5,999 (i) 250 × 8, 400 × 5, or 500 × 4 = 2,000
Q2: 25 × 31 → 750 · 132 × 19 → 2,600 · 101 × 11 → 1,000 (about 100 × 10) · 248 × 49 → 12,500 · 12 × 25 → 300
Page 89 · The King’s Reward

Who gets the most gold?

✅ AnswerMultiplying 7 times: Minister 1: 5 → 640 coins. Minister 2: 3 → 6,561 coins. Minister 3: 1 → 78,125 coins. Minister 3 wins, even though he started with just 1 coin!
(If the first day counts as the starting day, there are 6 multiplications: 320, 2,187, and 15,625. Minister 3 still wins.)
Pages 89–90 · Multiplication patterns

Predict the products

✅ Answer a) 3) 16 × 22 = 352 (half of 704) 4) 32 × 44 = 1,408 (double of 704)
b) 1) 6 × 16 = 96 2) 24 × 16 = 384 3) 24 × 64 = 1,536 4) 12 × 16 = 192
Q2: 1,111 × 1,111 = 12,34,321 · 45 × 45 = 2,025 · 55 × 55 = 3,025 · 11 × 54 = 594 · 11 × 82 = 902 · 1,234 × 9 + 4 = 11,110
Number pairs: smallest product 11 × 14 = 154, largest 35 × 75 = 2,625. Look for the smallest and biggest neighbours; you don’t need to find every product.
Page 91 · Let Us Solve

Final problems

✅ Answer 1. 18 books = 6 sets of 3 → 6 × 150 = 900. Then 900 + 20 = ₹920
2. (a) 57 × 115 = 6,555 and 3 × 1,599 = 4,797. Total = ₹11,352 (b) 1,750 + 1,129 = ₹2,879
3. (a) 12 + 17 + 1 = 30 rows (b) 18 + 22 + 1 = 41 students (counting Ananya) (c) 30 × 41 = 1,230 students
4. (a) 5,226 (b) 1,904 (c) 11,835 (d) 46,612 (e) 46,224 (f) 37,200
5. 67 × 68 = 4,489 + 67 = 4,556
6. 99 × 99 = 9,900 − 99 = 9,801

👨‍👩‍👧 Tips for Parents

  • Use the milk bill: “₹58 a litre, 2 litres a day, 30 days. What’s the monthly bill?”
  • Ask your child to show two different methods for the same sum and check that they match.
  • Practise × 10 and × 100 orally during short car or bus rides.
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