Multiplication and Division — Large Numbers
1. Multiplication — Introduction
Multiplication is REPEATED ADDITION.
'3 × 4 means 3 groups of 4, or 4 + 4 + 4 = 12.'
Key Terms:
- Multiplicand — The number being multiplied.
- Multiplier — The number you multiply by.
- Product — The answer.
Example: 345 × 6
- Multiplicand = 345
- Multiplier = 6
- Product = 2,070
2. Multiplication by 1-Digit Numbers
Steps:
- Multiply the ONES place first.
- CARRY if needed (just like addition).
- Multiply tens, hundreds, thousands — adding any carry each time.
Example:
Multiply 3,456 × 7
| Th | H | T | O | |
|---|---|---|---|---|
| 3 | 4 | 5 | 6 | |
| × | 7 | |||
| Carry | 2 | 3 | 4 | |
| Product | 2 | 4 | 1 | 9 |
Step 1: 7 × 6 = 42 → write 2, carry 4 Step 2: 7 × 5 = 35 + 4 (carry) = 39 → write 9, carry 3 Step 3: 7 × 4 = 28 + 3 (carry) = 31 → write 1, carry 3 Step 4: 7 × 3 = 21 + 3 (carry) = 24 → write 24
Answer: 24,192
3. Multiplication by 2-Digit Numbers
Steps:
- Multiply by the ONES digit of the multiplier. Write the PARTIAL PRODUCT.
- Put a ZERO in the ones place (or leave a blank) for the TENS digit multiplication.
- Multiply by the TENS digit. Write the second partial product.
- ADD both partial products.
Example:
Multiply 345 × 23
345
× 23
--------
1035 (345 × 3)
+ 6900 (345 × 20 — note the zero at the end)
--------
7935
Answer: 7,935
'When multiplying by tens digit, always put a ZERO in the ones place before multiplying.'
4. Multiplication by 3-Digit Numbers
Steps:
Same pattern — multiply by ones, then tens, then hundreds. Add a zero for each new row.
Example:
Multiply 234 × 123
234
× 123
--------
702 (234 × 3)
4680 (234 × 20)
+ 23400 (234 × 100)
--------
28782
Answer: 28,782
5. Properties of Multiplication
| Property | Meaning | Example |
|---|---|---|
| Commutative | Order does NOT matter. | 5 × 8 = 8 × 5 |
| Associative | Grouping does NOT matter. | (2 × 3) × 4 = 2 × (3 × 4) |
| Identity | Multiplying by 1 gives same number. | 567 × 1 = 567 |
| Zero Property | Multiplying by 0 gives 0. | 789 × 0 = 0 |
| Distributive | a × (b + c) = (a × b) + (a × c) | 3 × (4 + 5) = (3×4) + (3×5) |
Key Fact:
'Multiplication is COMMUTATIVE — 7 × 8 and 8 × 7 both equal 56. But division is NOT commutative!'
6. Division — Introduction
Division is SHARING EQUALLY or GROUPING.
'12 ÷ 3 means sharing 12 into 3 equal groups. Each group gets 4.'
Key Terms:
- Dividend — The number being divided.
- Divisor — The number you divide by.
- Quotient — The answer.
- Remainder — What is left over.
Formula: Dividend = Divisor × Quotient + Remainder
Important: 'The remainder is ALWAYS less than the divisor.'
7. Division by 1-Digit Numbers
Steps (Long Division):
- Start from the LEFTMOST digit.
- Divide → Multiply → Subtract → Bring down.
- Repeat until all digits are used.
Example:
Divide 8,456 ÷ 4
| 2 | 1 | 1 | 4 | |
|---|---|---|---|---|
| 4 | 8 | 4 | 5 | 6 |
| -8 | ||||
| 0 | 4 | |||
| -4 | ||||
| 0 | 5 | |||
| -4 | ||||
| 1 | 6 | |||
| -1 | ||||
| 0 |
Step 1: 4 ÷ 4 = 1 → 1 × 4 = 4, remainder 0 Step 2: Bring down 5. 5 ÷ 4 = 1 → 1 × 4 = 4, remainder 1 Step 3: Bring down 6. 16 ÷ 4 = 4 → 4 × 4 = 16, remainder 0
Answer: 2,114 Remainder 0
8. Division by 2-Digit Numbers
Example:
Divide 4,896 ÷ 12
| 4 | 0 | 8 | |
|---|---|---|---|
| 12 | 4 | 8 | 9 |
| -4 | 8 | ||
| 0 | 0 | 9 | |
| -0 | |||
| 9 | |||
| -9 | |||
| 0 |
Step 1: 48 ÷ 12 = 4 → 4 × 12 = 48, remainder 0 Step 2: Bring down 9. 9 ÷ 12 = 0 → 0 × 12 = 0, remainder 9 Step 3: Bring down 6. 96 ÷ 12 = 8 → 8 × 12 = 96, remainder 0
Answer: 408 Remainder 0
9. Word Problems
Problem 1 — Multiplication:
A book costs Rs. 345. What is the cost of 24 such books?
Solution: 345 × 24 = 345 × 20 + 345 × 4 = 6,900 + 1,380 = Rs. 8,280
Problem 2 — Division:
4,560 pencils are to be packed equally in 12 boxes. How many pencils in each box?
Solution: 4,560 ÷ 12 = 380 pencils per box
Problem 3 — Mixed:
A factory produces 2,345 toys each day. How many toys does it produce in 30 days? If these toys are packed in boxes of 15, how many boxes are needed?
Solution: 2,345 × 30 = 70,350 toys. 70,350 ÷ 15 = 4,690 boxes.
10. Common Mistakes
- Wrong placement of partial products: 'When multiplying by the tens digit, the first digit goes in the TENS place — always add a zero at the end.'
- Forgetting the remainder: 'After dividing, check: Remainder must be LESS than divisor. If remainder ≥ divisor, your quotient is too small.'
- Multiplication table errors: 'Know your tables up to 10×10 cold. A mistake in tables leads to a wrong answer.'
- Dividing from the right: 'In division, you start from the LEFT. In addition, subtraction, and multiplication, you start from the RIGHT. Don't mix them up!'
11. Key Facts to Remember
- 'Any number × 1 = the same number. Any number × 0 = 0.'
- 'Any number ÷ 1 = the same number.'
- 'You CANNOT divide by 0. It is undefined.'
- 'The remainder is ALWAYS less than the divisor.'
- 'Multiplication and division are INVERSE operations — they undo each other.'
12. Self-Test
Q1: Multiply: 5,678 × 9
Q2: Multiply: 456 × 34
Q3: Divide: 7,245 ÷ 5
Q4: Divide: 8,424 ÷ 12
Q5: A school collects Rs. 1,250 from each of 35 students for a picnic. How much money is collected in total?
Q6: 3,600 books are arranged equally on 24 shelves. How many books on each shelf?
Q7: Is 7,892 × 0 = 7,892? Why or why not?
Q8: A truck carries 1,560 kg of rice. If each bag weighs 12 kg, how many bags are there?
Answers:
A1: 51,102 A2: 15,504 A3: 1,449 A4: 702 A5: Rs. 43,750 A6: 150 books A7: No! Any number × 0 = 0, so 7,892 × 0 = 0. A8: 1,560 ÷ 12 = 130 bags
