NCERT Class 4 Maths · Chapter 9
Equal Groups
Multiplication and division made easy: skip jumps, multiples, arrays, doubling, the times-table chart, and multiplying and dividing using 10s and 100s.
📌 What you’ll learn in this chapter
- Multiples and common multiples (animal jumps)
- Arrays: rows × columns
- Doubling and patterns in the times-table chart
- Multiplying by 10s and 100s, and breaking numbers (18 × 4 = 10 × 4 + 8 × 4)
- Division as sharing and as making groups, with remainders
🐸 Let’s Begin: Hop, Hop, Hop!
A frog starts at 0 and jumps 5 steps each time: 5, 10, 15, 20, 25… These landing numbers are the multiples of 5. After 6 jumps it is at 6 × 5 = 30.
Multiplication is just equal jumps or equal groups!
🔢 Part 1: Multiples and Common Multiples
- Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24…
- Multiples of 4: 4, 8, 12, 16, 20, 24…
Numbers in both lists (12, 24, 36…) are common multiples of 3 and 4.
💡 Is 67 a multiple of 3?
Divide: 67 ÷ 3 = 22, remainder 1. There’s a remainder, so no. The frog lands on 66 and then 69, skipping 67.
🟦 Part 2: Arrays
Things arranged in rows and columns form an array. 4 rows of 6 chairs = 4 × 6 = 24 chairs. Turn the picture sideways and it’s 6 × 4 = 24. Same answer!
24 can be arranged as: 1 × 24, 2 × 12, 3 × 8, 4 × 6 (and the other way round).
✨ Part 3: Doubling Magic
Doubling means × 2. Double 34 → 30 + 30 and 4 + 4 → 60 + 8 = 68.
- When you double any number, the answer is always even.
- So the ones digit after doubling can only be 0, 2, 4, 6, or 8. Never 1, 3, 5, 7, or 9!
📊 Part 4: The Times-Table Chart
- Row 7 has the same numbers as column 7, because 7 × 4 = 4 × 7.
- Even rows (2, 4, 6, 8, 10) have only even products.
- Odd rows have both odd and even products, alternating.
- Row 5 ones digits go 5, 0, 5, 0 … and row 10 always ends in 0.
- Rows 1, 3, 7, and 9 have all ten digits (0–9) in the ones place, if you go far enough.
🔟 Part 5: Multiplying with 10s and 100s
6 × 8 = 48 → 60 × 8 = 480 (10 times more) → 600 × 8 = 4,800 (100 times more)
Break the number into parts
| × | 7 |
|---|---|
| 100 | 700 |
| 30 | 210 |
| 6 | 42 |
| 136 × 7 | 952 |
So 136 × 7 = 700 + 210 + 42 = 952.
➗ Part 6: Division
Division answers “How many in each group?” (sharing) or “How many groups?” (grouping).
Example: 75 bangles are packed in sets of 6. How many sets?
Take away 10 sets (60) → 15 left. Take away 2 sets (12) → 3 left.
10 + 2 = 12 sets, with 3 bangles left over (remainder).
Check: 12 × 6 + 3 = 75 ✅
Patterns: 8 ÷ 4 = 2 → 80 ÷ 4 = 20 → 800 ÷ 4 = 200
⚠️ Common mistake
Forgetting that the leftover must be smaller than the group size. If 7 are left over when packing in 6s, you can make one more set!
✏️ Practice Time
Try these on your own first, then check your answer below each question.
Q1. Write the first 5 multiples of 7.
✅ Answer7, 14, 21, 28, 35
Q2. 40 × 6 = ?
✅ Answer240
Q3. 24 × 5 = ? (break it: 20 × 5 + 4 × 5)
✅ Answer100 + 20 = 120
Q4. 96 ÷ 8 = ?
✅ Answer12
Q5. 50 children sit 6 to a bench. How many benches are needed?
✅ Answer50 ÷ 6 = 8, remainder 2, so 9 benches (the last bench has 2 children).
📘 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!
Pages 128–130 · Animal JumpsMultiples and common multiples
✅ Answer
1. The frog touches 3, 6, 9 … It will not touch 67 (66 and 69 are multiples of 3).
2. The squirrel touches multiples of 4. It needs 15 jumps to reach 60.
3. The rabbit touches multiples of 6. The smallest 3-digit number it lands on is 102, after 17 jumps.
4. The kangaroo touches multiples of 8. Both the rabbit and the kangaroo touch 24, 48, 72, 96…
5. To 48: the rabbit takes 8 jumps, and the kangaroo takes 6. Bigger jumps need fewer jumps.
6. To 60: the frog takes 20 jumps, and the rabbit takes 10. The rabbit’s jump is double, so it needs half as many.
Common multiples of 3 and 4: 12, 24, 36, 48 · of 6 and 8: 24, 48, 72, 96
7. Yes, the cat catches the rat at 24 (after 6 jumps each).
8. Look for three numbers next to each other in a line that make a fact, e.g. 3 × 4 = 12, 2 × 3 = 6, 10 ÷ 5 = 2, 16 ÷ 8 = 2, 2 × 7 = 14.
Pages 131–133 · Gulabo’s Garden2. The squirrel touches multiples of 4. It needs 15 jumps to reach 60.
3. The rabbit touches multiples of 6. The smallest 3-digit number it lands on is 102, after 17 jumps.
4. The kangaroo touches multiples of 8. Both the rabbit and the kangaroo touch 24, 48, 72, 96…
5. To 48: the rabbit takes 8 jumps, and the kangaroo takes 6. Bigger jumps need fewer jumps.
6. To 60: the frog takes 20 jumps, and the rabbit takes 10. The rabbit’s jump is double, so it needs half as many.
Common multiples of 3 and 4: 12, 24, 36, 48 · of 6 and 8: 24, 48, 72, 96
7. Yes, the cat catches the rat at 24 (after 6 jumps each).
8. Look for three numbers next to each other in a line that make a fact, e.g. 3 × 4 = 12, 2 × 3 = 6, 10 ÷ 5 = 2, 16 ÷ 8 = 2, 2 × 7 = 14.
Petals, saplings, and cupcakes
✅ Answer1. 2 lilies = 6 petals, and 12 lilies = 36 petals.
2. 80 petals = 16 flowers.
3. Count the saplings in one row and the number of rows, then multiply (rows × saplings in each row).
4. Count the columns of strawberry boxes and the boxes in each column, then multiply (it looks like 16 × 6 = 96; check in your book).
5. (a) 18 cupcakes = 3 rows of 6 in each tray (b) Two trays = 36 (c) 108 ÷ 18 = 6 trays (d) 36 = 6 × 6 in a square tray
Other arrangements: 36 = 4 × 9, 3 × 12 · 8 = 2 × 4 · 12 = 3 × 4, 2 × 6 · 24 = 4 × 6, 3 × 8
Page 134 · Doubling Magic2. 80 petals = 16 flowers.
3. Count the saplings in one row and the number of rows, then multiply (rows × saplings in each row).
4. Count the columns of strawberry boxes and the boxes in each column, then multiply (it looks like 16 × 6 = 96; check in your book).
5. (a) 18 cupcakes = 3 rows of 6 in each tray (b) Two trays = 36 (c) 108 ÷ 18 = 6 trays (d) 36 = 6 × 6 in a square tray
Other arrangements: 36 = 4 × 9, 3 × 12 · 8 = 2 × 4 · 12 = 3 × 4, 2 × 6 · 24 = 4 × 6, 3 × 8
Anvi’s magic
✅ AnswerAnvi doubles the flowers: 23 → 46. The table: 10 → 20 · 51 → 102 · 95 → 190 · 150 → 300 · 199 → 398 · 425 → 850 · 500 → 1,000 · and 111 → 222 · 205 → 410 · 250 → 500
(a) 64 (b) 28 (c) 52 (d) 34 (e) 78 (f) 90
Ones digits after doubling: 28 → 6 · 56 → 2 · 45 → 0 · 17 → 4
To get 0: 5, 10 · 2: 1, 6 · 4: 2, 7 · 6: 3, 8 · 8: 4, 9. We cannot get 3, 5, 7, or 9, because doubled numbers are always even.
Pages 135–136 · Times-table chart(a) 64 (b) 28 (c) 52 (d) 34 (e) 78 (f) 90
Ones digits after doubling: 28 → 6 · 56 → 2 · 45 → 0 · 17 → 4
To get 0: 5, 10 · 2: 1, 6 · 4: 2, 7 · 6: 3, 8 · 8: 4, 9. We cannot get 3, 5, 7, or 9, because doubled numbers are always even.
Patterns
✅ AnswerThe green diagonal shows 1, 4, 9, 16, 25… (a number times itself). Row 7 = column 7, because the order doesn’t matter in multiplication. Rows 2, 4, 6, 8, and 10 have all even products. No row has only odd products. Odd rows have both, because odd × even = even and odd × odd = odd. There are more even numbers (75 even, 25 odd).
Common multiples: 2 and 3 → 6, 12, 18… · 4 and 8 → the multiples of 8 · 7 and 9 → 63.
Row 5: 5, 0, 5, 0… · Row 8: 11 × 8 → 8, 12 × 8 → 6, 13 × 8 → 4
Row 8 never ends in 1, 3, 5, 7, or 9. Rows with all digits 0–9 in the ones place: 1, 3, 7, 9. __ × 8 ends in 0 for 5, 10, 15, 20, 25… Rows where 0 appears only once (up to × 10): 1, 3, 7, 9. These are the same rows as in Q11!
Pages 137–139 · Multiples of tensCommon multiples: 2 and 3 → 6, 12, 18… · 4 and 8 → the multiples of 8 · 7 and 9 → 63.
Row 5: 5, 0, 5, 0… · Row 8: 11 × 8 → 8, 12 × 8 → 6, 13 × 8 → 4
Row 8 never ends in 1, 3, 5, 7, or 9. Rows with all digits 0–9 in the ones place: 1, 3, 7, 9. __ × 8 ends in 0 for 5, 10, 15, 20, 25… Rows where 0 appears only once (up to × 10): 1, 3, 7, 9. These are the same rows as in Q11!
Wheels, cupcakes, and rods
✅ Answer10 × 3 = 30 · 20 × 3 = 30 + 30 = 60 · 10 × 4 = 40 · 30 × 4 = 40 + 40 + 40 = 120
(a) 60 (b) 240 (c) 80 (d) 480 (e) 48 (f) 480 (g) 24 (h) 320
Cupcakes: 40 + 32 = 72 · Rods: 160 + 24 = 184
Let Us Solve: 1. 25 × 2 + 12 × 4 = 50 + 48 = 98 legs 2. 120 3. 126 4. Count the rows and columns of beads, multiply, then count each colour. 5. (a) 102 (b) 375 (c) 276 (d) 450
Pages 140–141 · Division(a) 60 (b) 240 (c) 80 (d) 480 (e) 48 (f) 480 (g) 24 (h) 320
Cupcakes: 40 + 32 = 72 · Rods: 160 + 24 = 184
Let Us Solve: 1. 25 × 2 + 12 × 4 = 50 + 48 = 98 legs 2. 120 3. 126 4. Count the rows and columns of beads, multiply, then count each colour. 5. (a) 102 (b) 375 (c) 276 (d) 450
Tempos, cows, and octopuses
✅ AnswerTempos: 28 left → 15 for 5 tempos → 13 left → 9 for 3 tempos → 4 left → 3 for 1 tempo → 1 left. No more tempos can be made. Total: 19 tempos.
Cows: 4 legs each. 88 ÷ 4 = 10 + 10 + 2 = 22 cows.
1. 72 ÷ 8 = 9 octopuses 2. 50 ÷ 4 = 12 boxes with 2 left over. They need 13 boxes to pack all (the last box is not full). 3. 15 rows 4. (a) 14 (b) 12 (c) 23 (d) 15 remainder 3
Pages 142–145 · Multiples of 100Cows: 4 legs each. 88 ÷ 4 = 10 + 10 + 2 = 22 cows.
1. 72 ÷ 8 = 9 octopuses 2. 50 ÷ 4 = 12 boxes with 2 left over. They need 13 boxes to pack all (the last box is not full). 3. 15 rows 4. (a) 14 (b) 12 (c) 23 (d) 15 remainder 3
Bikes, cars, autos, and kulhads
✅ Answer100 × 2 = 200 · 200 bikes → 400 · 100 × 4 = 400 · 500 cars → 2,000 · 5 × 4 = 20, 50 × 4 = 200
Pattern: 3, 30, 300 · 6, 60, 600 · 12, 120, 1,200 · 8, 80, 800 (for each of the three columns) · 12, 120, 1,200 · 15, 150, 1,500 · 27, 270, 2,700
Autos: 800 + 160 + 40 = 1,000 · Kulhads: 600 + 420 + 24 = 1,044
Let Us Solve: 1. 465 × 2 = 930 pencils 2. 234 × 5 = ₹1,170 3. (a) 1,756 (b) 4,112 (c) 1,780 (d) 4,361
Page 146 · Patterns in divisionPattern: 3, 30, 300 · 6, 60, 600 · 12, 120, 1,200 · 8, 80, 800 (for each of the three columns) · 12, 120, 1,200 · 15, 150, 1,500 · 27, 270, 2,700
Autos: 800 + 160 + 40 = 1,000 · Kulhads: 600 + 420 + 24 = 1,044
Let Us Solve: 1. 465 × 2 = 930 pencils 2. 234 × 5 = ₹1,170 3. (a) 1,756 (b) 4,112 (c) 1,780 (d) 4,361
Sharing money
✅ Answer₹30 ÷ 3 = ₹10 each · ₹900 ÷ 3 = ₹300 each
A: 1, 10, 100 · 1, 10, 100 · B: 3, 30, 300 · 2, 20, 200 · C: 5, 50, 100 · 2, 10, 50
1. 60 ÷ 6 = 10 trucks 2. 80 ÷ 4 = 20 cars 3. 600 ÷ 6 = 100 ants 4. 800 ÷ 4 = 200 packets
Pages 147–148 · Let Us SolveA: 1, 10, 100 · 1, 10, 100 · B: 3, 30, 300 · 2, 20, 200 · C: 5, 50, 100 · 2, 10, 50
1. 60 ÷ 6 = 10 trucks 2. 80 ÷ 4 = 20 cars 3. 600 ÷ 6 = 100 ants 4. 800 ÷ 4 = 200 packets
Buses, toy train, and rafts
✅ Answer1. 245 ÷ 7 = 35 children per bus 2. 88 × 7 = 616 km 3. 259 ÷ 7 = 37 rafts 4. (a) ₹270 (b) ₹45 each (c) ₹90 each
5. Only (b) 56 trips and (d) 224 trips can be worked out. We don’t know the fare or the time of a trip, so (a) and (c) can’t be answered.
6. (a) 405 (b) 3,549 (c) 23 R 2 (d) 129 R 4 (e) 470 (f) 1,584 (g) 27 R 2 (h) 127
7. (a) Always (b) Never (c) Sometimes (5 × 3 = 15, but 5 × 2 = 10) (d) Always (e) Sometimes (f) Never
5. Only (b) 56 trips and (d) 224 trips can be worked out. We don’t know the fare or the time of a trip, so (a) and (c) can’t be answered.
6. (a) 405 (b) 3,549 (c) 23 R 2 (d) 129 R 4 (e) 470 (f) 1,584 (g) 27 R 2 (h) 127
7. (a) Always (b) Never (c) Sometimes (5 × 3 = 15, but 5 × 2 = 10) (d) Always (e) Sometimes (f) Never
👨👩👧 Tips for Parents
- Count things in rows (eggs, chairs, tiles) using multiplication.
- Share snacks equally and talk about remainders.
- Play a doubling game: you say a number, and your child doubles it.