Guided Long Multiplication

How does long multiplication work?

We start with the rightmost digit of the multiplier (the units digit) and multiply it by every digit of the multiplicand, from right to left. Whenever the product is greater than 9, we write down the units digit in the partial-product row, and carry the tens digit forward as a carry — adding it into the next multiplication.

Once we’ve finished the multiplier’s units digit, we move on to the tens digit. Since it represents tens, its partial product is written shifted one place to the left — that is, with one zero on the right. The multiplier’s hundreds digit will give a partial product with two zeros on the right, and so on. Finally, we add all the partial products in columns to get the result.

Three examples

Common mistakes

• Forgetting the zeros — multiplying by tens needs one zero on the right, hundreds needs two zeros, and so on. Forget the zeros and the result comes out far smaller than the true answer.
• A carry that gets dropped. Whenever a product is ≥10, its tens digit must be added into the next multiplication — not left hanging.
• Misaligned columns in the final addition.Units under units, tens under tens. Correct working in every row won’t help if the final sum is misaligned.
• A digit of 0 in the multiplier. A whole row of zeros — but you still need the correct shift so the next row lands in the right place.