Long Multiplication

Long multiplication breaks a multi-digit multiplication into smaller steps using partial products, then adds them together.

The Big Idea

When you multiply a large number by another large number, you can break it into partial products — multiply by each digit separately, then add the results together.

Long Multiplication

A \times B = A \times (b_1 + b_0) = A \cdot b_1 \cdot 10 + A \cdot b_0

Multiply A by each digit of B (adjusted for place value), then sum all partial products.

Multiplying by a Single Digit

1
Multiply the ones digit
Multiply the bottom single digit by the ones digit of the top number. Write the result (carry if ≥ 10).
2
Move left
Multiply by the tens digit of the top number, adding any carry. Continue until all digits are done.
3
Write the result
The digits form your answer.

Calculate 476 × 3

  476
×   3
-----

Ones: 3 × 6 = 18 → write 8, carry 1

Tens: 3 × 7 = 21 + 1 = 22 → write 2, carry 2

Hundreds: 3 × 4 = 12 + 2 = 14 → write 14

Result: 1 428

Multiplying by a Multi-Digit Number

1
Multiply by the ones digit of the multiplier
Write the partial product on the first line.
2
Multiply by the tens digit of the multiplier
Write this partial product on the second line, shifted one place to the left (or add a zero placeholder).
3
Continue for each digit
Each new partial product shifts one more place to the left.
4
Add all partial products
Use column addition to find the final answer.

Worked Example

Calculate 476 × 38

    476
×    38
-------

Step 1 — multiply by 8 (ones):

8 × 6 = 48 → write 8, carry 4

8 × 7 = 56 + 4 = 60 → write 0, carry 6

8 × 4 = 32 + 6 = 38 → write 38

Partial product 1: 3 808

Step 2 — multiply by 3 (tens), shift left by one:

3 × 6 = 18 → write 8, carry 1

3 × 7 = 21 + 1 = 22 → write 2, carry 2

3 × 4 = 12 + 2 = 14 → write 14

Partial product 2: 14 280 (shifted: 14 28 with a 0 at end)

Step 3 — add:

   3 808
+ 14 280
--------
  18 088

Result: 476 × 38 = 18 088

Tip Write a zero at the end of each new partial product line to keep your place values correct. The second line has one zero, the third has two zeros, and so on.

Real-World Applications

Real world: calculating total cost

A school orders 38 boxes of coloured pencils. Each box costs 476 cents (4,76 €). What is the total cost?

We already found 476 × 38 = 18 088 cents = 180,88 €

Real world: calculating area

A rectangular sports field measures 124 m by 65 m. What is its area?

    124
×    65
-------

Partial product (× 5): 620

Partial product (× 60): 7 440

Area: 620 + 7 440 = 8 060 m²

Real world: weekly earnings

A part-time worker earns 345 € per month. How much does she earn in 12 months?

    345
×    12
-------

Partial product (× 2): 690

Partial product (× 10): 3 450

Annual earnings: 690 + 3 450 = 4 140 €

Common Mistakes

Forgetting to shift partial products to the left for each successive digit
Not adding the carried digit before writing
Adding partial products without aligning them by place value

Practice Problems

Calculate 325 × 46.

Show answer

325 × 6 = 1 950. 325 × 40 = 13 000. Total: 1 950 + 13 000 = 14 950.

A factory produces 248 parts per day. How many parts are produced in 25 working days?

Show answer

248 × 25 = 248 × 20 + 248 × 5 = 4 960 + 1 240 = 6 200 parts.