Long Multiplication

What is long multiplication?

Long multiplication is a written method for multiplying numbers that are too big to do quickly in your head — for example 37 × 24 or 243 × 16. Instead of one giant calculation, you break the problem into small, easy steps using your times tables, then add the pieces together.

Before you start, you need to be confident with your times tables and comfortable with the idea of introduction to multiplication. If those feel solid, you are ready.

The big idea: break numbers apart

The whole trick of long multiplication comes from place value. The number 24 is really 20 + 4. So when we work out 37 × 24, we are really doing:

37 × 24 = (37 × 4) + (37 × 20)

We do the two easy multiplications separately and then add them. Those two answers are called the partial products. This works because of a rule called the distributive law — you can split one of the numbers up, multiply each part, and add the results without changing the answer.

Step-by-step method

Let's multiply 37 × 24 using the column method. Write the numbers one above the other, lining up the ones, tens and hundreds.

    37
  × 24
  ----

Step 1 — Multiply by the ones digit (4).

• 4 × 7 = 28. Write 8 in the ones column, carry the 2 (which is 2 tens) above the 3.
• 4 × 3 = 12, plus the carried 2 = 14. Write 14.
• First partial product: 148.

Step 2 — Multiply by the tens digit (2).

The 2 is really 20, so first write a 0 in the ones column to hold the place. Now multiply by 2:

• 2 × 7 = 14. Write 4, carry the 1.
• 2 × 3 = 6, plus the carried 1 = 7. Write 7.
• Second partial product: 740.

Step 3 — Add the partial products.

    37
  × 24
  ----
   148   ← 37 × 4
   740   ← 37 × 20
  ----
   888

So 37 × 24 = 888.

Why the zero placeholder?

This is the part that confuses many learners, so let's make it clear. When you multiply by the tens digit, you are multiplying by 20, not 2. An answer for "× 20" is always ten times bigger than the answer for "× 2", and in our number system making something ten times bigger means shifting every digit one place to the left. The 0 simply holds the empty ones place so the digits land in their correct columns. If you forget the zero, all your digits sit in the wrong place and the answer is far too small.

Worked example 2: three digits

Let's try 243 × 16.

Multiply by 6 (the ones):

• 6 × 3 = 18 → write 8, carry 1
• 6 × 4 = 24, + 1 = 25 → write 5, carry 2
• 6 × 2 = 12, + 2 = 14 → write 14
• First partial product: 1458

Multiply by 1 (the tens, really 10) — write a 0 first:

• 1 × 3 = 3, 1 × 4 = 4, 1 × 2 = 2
• Second partial product: 2430

Add them:

    243
  ×  16
  -----
   1458   ← 243 × 6
   2430   ← 243 × 10
  -----
   3888

So 243 × 16 = 3888.

A handy summary table

Stage	What you do	Example (37 × 24)
1	Multiply by the ones digit	37 × 4 = 148
2	Write a 0, then multiply by the tens digit	37 × 20 = 740
3	Add the partial products	148 + 740 = 888

Checking your answer

Always do a quick sanity check using rounding. For 37 × 24, round to 40 × 20 = 800. Our answer, 888, is close to 800 — that feels right. If you had got 8,880 or 88, you would know something went wrong. This rounding trick catches most mistakes, especially forgotten zeros.

Why long multiplication matters

You won't always have a calculator, and even when you do, understanding how multiplication works helps you spot when a calculator answer looks wrong. Long multiplication also builds the place-value thinking you'll need for algebra, area calculations, and money problems later. It teaches you to break a hard problem into small, manageable steps — a skill that helps far beyond math.

Try it yourself

Work these out using the column method. Show your partial products and remember the zero placeholder.

1. 26 × 13
2. 54 × 32
3. 18 × 45
4. 312 × 24 (a three-digit challenge!)

Self-check: estimate each answer first by rounding. Does your final answer land close to your estimate?

Answers: 1) 338 2) 1,728 3) 810 4) 7,488.

When you're ready for the reverse skill — splitting big numbers up — head over to our lesson on division made simple.