Multiplying bigger numbers
When both numbers have more than one digit, we use long multiplication. The idea: break one number into its place-value parts, multiply each part, then add the results. Those pieces are called partial products.
The area model
The area model shows why long multiplication works. For 34 × 26, split both numbers:
| × | 30 | 4 |
|---|---|---|
| 20 | 600 | 80 |
| 6 | 180 | 24 |
Add the four partial products: 600 + 80 + 180 + 24 = 884.
The standard method
3 4
× 2 6
-----
2 0 4 ← 34 × 6
+ 6 8 0 ← 34 × 20
-------
8 8 4
Step 1: Multiply by the ones digit (6).
- 6 × 4 = 24. Write 4, carry 2.
- 6 × 3 = 18, plus 2 = 20. Write 20.
- First row: 204.
Step 2: Multiply by the tens digit (2, which is really 20).
- Write a placeholder zero in the ones column first.
- 2 × 4 = 8. Write 8.
- 2 × 3 = 6. Write 6.
- Second row: 680.
Step 3: Add the rows: 204 + 680 = 884.
Three-digit numbers
For 312 × 45:
- 312 × 5 = 1,560
- 312 × 40 = 12,480 (placeholder zero)
- Total: 1,560 + 12,480 = 14,040
If the bottom number has three digits, you get three rows, with one zero for the tens row and two zeros for the hundreds row.
Estimate first
Estimate 34 × 26 ≈ 30 × 30 = 900. Our answer 884 is close ✓.
Worked examples
Example 1: 47 × 23
- 47 × 3 = 141
- 47 × 20 = 940
- Sum: 1,081
Example 2: 58 × 14
- 58 × 4 = 232
- 58 × 10 = 580
- Sum: 812
Example 3: 125 × 32
- 125 × 2 = 250
- 125 × 30 = 3,750
- Sum: 4,000
Common mistakes
- Forgetting the placeholder zero, so the second row lines up wrong and the answer is far too small.
- Adding the carried digit before multiplying instead of after.
- Mixing up carried digits from the first row with the second row. Cross out old carries.
- Not adding the partial products at the end.
Try it yourself
- 23 × 14
- 56 × 37
- 204 × 18
- 99 × 99
Answers: 1) 322 2) 2,072 3) 3,672 4) 9,801
Quick recap
- Long multiplication finds partial products for each digit and adds them.
- Use a placeholder zero when multiplying by tens.
- The area model shows why the method works; estimate to check.