How to do long multiplication

Long multiplication breaks a big multiplication into small ones you already know from the times tables. You multiply by one digit at a time, write each answer on its own row, and add the rows up.

The rule

Multiply the top number by each digit of the bottom number in turn, starting with the ones. Start each new row one place further to the left, then add the rows together.

How to do it

Write the longer number on top and the other underneath, lined up on the right, ones under ones and tens under tens.
Multiply the top number by the ones digit of the bottom number, working from right to left. Write the ones digit of each answer and carry the tens into the next column.
Move to the tens digit of the bottom number and start a new row. Write a 0 in the ones column first: you are multiplying by tens now, so nothing can land in the ones. Then multiply as before.
Do the same for every digit of the bottom number, starting one place further left each time.
Draw a line and add the rows. That sum is the answer.

Worked examples

What is 47 × 36?

			4	7
×			3	6
		2	8	2
+	1	4	1	0
	1	6	9	2

47 × 6 = 282, and 47 × 30 = 1410. Add the two rows, and 47 × 36 = 1692.

What is 213 × 124?

			2	1	3
×			1	2	4
			8	5	2
		4	2	6	0
+	2	1	3	0	0
	2	6	4	1	2

Three digits underneath, so three rows (852, 4260 and 21300). They add up to 213 × 124 = 26412.

What is 58 × 7?

		5	8
×			7
	4	0	6

A one-digit number underneath needs only one row, so there is nothing to add: 8 × 7 = 56, write 6 and carry 5; 5 × 7 = 35, plus the 5 is 40. 58 × 7 = 406.

The grid method, and why it is the same thing

Most British schools teach the grid method first. Split both numbers into tens and ones, multiply every part by every other part in a grid, and add the lot: 47 × 36 is 40 × 30 = 1200, 40 × 6 = 240, 7 × 30 = 210 and 7 × 6 = 42, which make 1692.

Long multiplication is the grid with the adding done as you go. The first row, 282, is 40 × 6 and 7 × 6 together; the second, 1410, is 40 × 30 and 7 × 30. Fewer numbers to write down, the same four multiplications underneath.

Short multiplication

When the bottom number has only one digit, as in 58 × 7, there is only one row and so nothing to add. That is called short multiplication, and it is what each row of a long multiplication is.

Why it works

36 is 30 + 6, so 47 × 36 is 47 × 6 and 47 × 30 added together. The first row is the 47 × 6. The second row is the 47 × 30, which is 47 × 3 shifted one place to the left, and the shift is all the 0 in the ones column is doing. Leave it out and the row says 141 when it means 1410.

A quick estimate catches most slips: 47 × 36 is a bit less than 50 × 40 = 2000, so 1692 is believable and 16920 or 169 are not.

Where mistakes happen

Forgetting to shift the second row. Every row after the first starts one place further left, with a 0 holding the empty places.
Adding the carry before multiplying. In 58 × 7, the 5 is multiplied by 7 first and the carried 5 is added after: 35 + 5 = 40, not (5 + 5) × 7.
Using an old carry on a new row. The carries from the first row are finished with once the row is done; cross them out before starting the next one.
Lining the numbers up on the left. They line up on the right, so that every column holds one place value.

Questions people ask

Does it matter which number goes on top?

No. 47 × 36 and 36 × 47 are the same answer. Putting the longer number on top means fewer rows to add.

Do I have to write the 0 at the start of the second row?

You need the space; the 0 just keeps it. Some American schools leave the place empty and start the row one column to the left instead, which comes to the same thing as long as the columns stay straight.

How do I check my answer?

Estimate first by rounding both numbers, and the answer should be close. For an exact check, divide the answer by one of the numbers. Long division takes you back to the other.

What about decimals?

The same method, plus one step at the end to place the point. See how to multiply decimals.