Long multiplication: how to do it, step by step

What is long multiplication?

Long multiplication is a way of multiplying by a two-digit number on paper, one digit at a time. It's called "long" because it takes more rows than short multiplication, which children use for multiplying by a one-digit number.

It works because of partitioning. To multiply by 23, you multiply by 3 and by 20 separately, then add the answers. That's the same idea as the grid method, set out in columns instead of a grid. If partitioning is new to your child, start there.

How to do long multiplication, step by step

To do long multiplication, write the bigger number on top and the two-digit number underneath, with the ones lined up. Then:

1. Multiply the top number by the ones digit and write the answer on the first row.
2. On the next row, write a 0 in the ones column. Then multiply the top number by the tens digit and write the answer to the left of the 0.
3. Add the two rows with column addition.

Here's a two-digit number multiplied by a two-digit number.

1. Multiply by the 3 ones: 46 × 3 = 138.
2. Multiply by the 2 tens, which is × 20: write a 0 in the ones column first as a placeholder, then 46 × 2 = 92, so 46 × 20 = 920.
3. Add the rows: 138 + 920 = 1,058.

Your child's school may set it out a little differently, for example with a plus sign beside the rows or the carried digits in another place. The national curriculum gives examples of the layout and says the exact position of the small carried digits varies. Follow the way your child has been taught.

Why the placeholder zero matters

The placeholder zero is there because the second row isn't multiplying by 2, it's multiplying by 20. In 23, the 2 is in the tens place, so it's worth 2 tens. Multiplying by 20 is the same as multiplying by 2 and then by 10, and multiplying by 10 moves every digit one place to the left. That leaves the ones column empty, and the 0 holds its place.

Without it, the row for 46 × 20 would say 92 instead of 920, and the final answer would come out far too small. Ask your child "what is the 2 really worth?" and the zero makes sense. Our guide to place value explains why digits move when you multiply by 10.

Long multiplication with bigger numbers

The method is exactly the same for bigger top numbers. There are still only two rows, because the bottom number still has two digits. The rows are just longer, and there are more digits to carry.

Three digits by two digits

1. Multiply by the 6 ones: 243 × 6 = 1,458.
2. Multiply by the 3 tens, which is × 30: write a 0 in the ones column first as a placeholder, then 243 × 3 = 729, so 243 × 30 = 7,290.
3. Add the rows: 1,458 + 7,290 = 8,748.

Four digits by two digits

This is the biggest calculation the curriculum asks for in Year 6: up to 4 digits by a two-digit number.

1. Multiply by the 7 ones: 1,342 × 7 = 9,394.
2. Multiply by the 2 tens, which is × 20: write a 0 in the ones column first as a placeholder, then 1,342 × 2 = 2,684, so 1,342 × 20 = 26,840.
3. Add the rows: 9,394 + 26,840 = 36,234.

Estimate first, then check

Before working it out, round both numbers to get a rough answer. The curriculum asks children to use rounding to check answers in Year 5, and estimation in Year 6.

Calculation	Estimate	Exact answer
46 × 23	50 × 20 = 1,000	1,058
243 × 36	200 × 40 = 8,000	8,748
1,342 × 27	1,000 × 30 = 30,000	36,234

The estimate won't match the answer, and it doesn't need to. It's there to catch big slips. For 46 × 23, an answer of 1,058 is close to 1,000, so it's believable. An answer of 230 is far too small, and the usual cause is a missing placeholder zero.

Long multiplication with decimals

In Year 6, children multiply numbers with up to two decimal places by one-digit and two-digit whole numbers. The simplest way is to take the decimal point out, do the long multiplication with whole numbers, then put the point back.

For 2.43 × 36, think of 2.43 as 243 hundredths. Work out 243 × 36 with long multiplication: it's 8,748, as in the example above. So 2.43 × 36 is 8,748 hundredths, which is 87.48. An estimate checks the decimal point is in the right place: 2 × 40 = 80, so 87.48 is sensible and 874.8 is not.

Long multiplication in Year 5 and Year 6

In Year 5, children multiply numbers with up to 4 digits by a one-digit or two-digit number using a formal written method, including long multiplication for two-digit numbers. In Year 6 they keep practising it with numbers up to 4 digits, and the curriculum expects them to be fluent in written methods for all four operations by the end of the year. See our Year 5 maths and Year 6 maths guides for what else comes in those years.

Every step depends on the times tables. Government guidance for teachers says the facts up to 9 × 9 are the ones needed for written multiplication, so it's worth keeping them quick with the times tables test or the multiplication grid.

Long multiplication in the KS2 SATs

Long multiplication comes up in Paper 1, the arithmetic paper, of the Year 6 SATs. The KS2 mathematics test framework says most arithmetic questions are worth one mark, but two marks are available for long multiplication. If the final answer is wrong, a mark can still be given for correct working using the formal method. Partial marks aren't given where the grid method has been used for long multiplication, so it's worth your child showing the long multiplication layout. Our KS2 maths SATs guide has the dates and papers, and our free SATs arithmetic practice paper has two long multiplication questions to try.

Common long multiplication mistakes

• Forgetting the placeholder zero. The second row comes out ten times too small. Ask what the tens digit is really worth.
• Columns drifting. Digits that aren't lined up get added to the wrong place. Squared paper, one digit per square, helps a lot.
• Mixing up carried digits. Carries from the first row get added into the second row by mistake. Many children cross out or rub out their carries once a row is done.
• Times table slips. One wrong fact, like thinking seven eights are 54, spoils the whole answer. Check any fact that took a while to come.
• Adding the rows wrongly. The final step is column addition, and it needs the same care as the multiplying.
• Multiplying the tens digit by the tens digit only. In 46 × 23, the 2 multiplies the whole of 46, not just the 4.

How to help at home

• Ask for an estimate first. Make it a habit before every written calculation.
• Check with the inverse. Once your child knows division, they can check an answer by dividing it back.
• Talk about the zero. Ask "why is that 0 there?" until the answer about tens comes easily.
• Keep the tables fresh. A few minutes most days of mixed times tables makes long multiplication much less tiring. Our guide on how to learn times tables has a routine.
• Use squared paper. It keeps the columns straight.
• Follow the school's layout. Ask your child to show you how they set it out in class.

Long multiplication questions to try

1. 34 × 12
2. 57 × 24
3. 125 × 32
4. 408 × 15
5. 2,316 × 23
6. 3.14 × 21

Answers

1. 408
2. 1,368
3. 4,000
4. 6,120
5. 53,268
6. 65.94 (314 × 21 = 6,594, then put the point back: 65.94)

Once long multiplication is secure, the next written method is long division, which uses the same times tables and the same care with columns.