Long Multiplication
Master multiplying multi-digit numbers using the long multiplication method. Break big problems into smaller, manageable steps.
How Long Multiplication Works
Long multiplication lets you multiply large numbers by breaking them into partial products. You multiply by each digit of the bottom number separately, then add the results together.
Example: 346 × 23
3 4 6
× 2 3
1 0 3 8 ← 346 × 3
6 9 2 0 ← 346 × 20
7 9 5 8 ← add together
Step 1
Multiply by the ones digit (3)
Step 2
Multiply by the tens digit (20) — add a zero
Step 3
Add the partial products together
Carrying (Regrouping)
When a multiplication gives a two-digit answer, write the ones digit in the answer and carry the tens digit to the next column. Remember to add the carried number after multiplying.
Example: 346 × 3 (the ones row)
Step 1: 3 × 6 = 18. Write 8, carry 1.
Step 2: 3 × 4 = 12, plus 1 carried = 13. Write 3, carry 1.
Step 3: 3 × 3 = 9, plus 1 carried = 10. Write 10.
Result: 1 038
Real-World Applications
Long multiplication is useful whenever you need to work with larger quantities in everyday life.
Shopping
A school orders 24 packs of exercise books at $18 each. What is the total cost?
24 × 18 = $432
Distance
A car travels 135 km each day for 12 days. How far does it travel in total?
135 × 12 = 1 620 km
Estimation and Checking
Always estimate first to check if your answer is reasonable. Round each number to the nearest ten or hundred, then multiply.
Example: Check 346 × 23 = 7 958
Estimate: 350 × 20 = 7 000
Our answer of 7 958 is close to 7 000, so it looks reasonable.
Key Vocabulary
Partial Product
The result of multiplying by one digit of the bottom number. You add partial products to get the final answer.
Carrying (Regrouping)
Moving the tens digit to the next column when a multiplication gives a two-digit result.
Factor
The numbers being multiplied together (e.g. in 346 × 23, both 346 and 23 are factors).
Product
The answer to a multiplication (e.g. 346 × 23 = 7 958. The product is 7 958).
Worked Examples
Calculate 254 × 16
Step 1: 254 × 6 = 1 524
Step 2: 254 × 10 = 2 540
Step 3: 1 524 + 2 540 = 4 064
A cinema has 28 rows with 36 seats in each row. How many seats are there?
Step 1: 36 × 8 = 288
Step 2: 36 × 20 = 720
Step 3: 288 + 720 = 1 008 seats
Calculate 1 205 × 14
Step 1: 1 205 × 4 = 4 820
Step 2: 1 205 × 10 = 12 050
Step 3: 4 820 + 12 050 = 16 870
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
What is 45 × 32?
Question 2
What is 123 × 15?
Question 3
A farmer plants 48 rows of tomatoes with 25 plants in each row. How many tomato plants are there?
Question 4
What is the first partial product when calculating 267 × 34?
Question 5
Which estimate is closest for 389 × 21?
Key Concepts Summary
- ●Long multiplication breaks a problem into partial products.
- ●Multiply by each digit of the bottom number, starting from the ones.
- ●When multiplying by the tens digit, add a zero placeholder.
- ●Carry (regroup) when a column product exceeds 9.
- ●Always estimate first to check your answer is reasonable.