How to Do Long Multiplication

Line the numbers up on the right and work from right to left, one digit at a time. When a product is 10 or more, write the ones digit and carry the tens to the next column. Here is 478 × 6:

1. 8 × 6 = 48: write 8, carry 4.
2. 7 × 6 = 42, plus the carried 4 is 46: write 6, carry 4.
3. 4 × 6 = 24, plus 4 is 28. This is the last digit, so write all of 28.

Always multiply first and add the carry second. Adding the carry before multiplying is one of the most common slips.

With two digits on the bottom you make two rows, one for each digit, and add them. In 478 × 36:

1. First row, the ones digit: 478 × 6 = 2,868.
2. Second row, the tens digit. The 3 means 30, so write a placeholder 0 on the right, then multiply by 3: 478 × 3 = 1,434, giving 14,340.
3. Add the rows: 2,868 + 14,340 = 17,208.

Each new row starts one more place to the left: one zero for tens, two for hundreds, three for thousands.

A 0 in the bottom number gives a row of all zeros, so you can skip it. Just make sure the next row still gets the right number of placeholder zeros. In 1,205 × 304 the tens row is skipped, and the hundreds row starts with two zeros:

A 0 in the top number is easier still: 0 × 4 = 0, plus any carry. In the first row, 5 × 4 = 20 (write 0, carry 2), then 0 × 4 + 2 = 2.

Many schools teach the box method first because it shows why long multiplication works. Split each number into place values, multiply every pair, and add the boxes. For 23 × 14, with 23 = 20 + 3 and 14 = 10 + 4:

The row totals (230 and 92) are exactly the rows you'd write in long multiplication, so the two methods always agree. The box method takes more space but has no carrying to track.

Unlike adding decimals, you don't line up the decimal points. Ignore them, multiply as whole numbers, then count the decimal places in both numbers and put that many in the answer. For 4.78 × 3.6, multiply 478 × 36 = 17,208. There are 2 + 1 = 3 decimal places, so the answer is 17.208.

Count the places before dropping zeros at the end: 2.5 × 0.4 is 25 × 4 = 100 with 2 places, so 1.00, which is 1.

Multiply without the signs, then decide the sign of the answer:

• One negative: the answer is negative. −12 × 15 = −180
• Two negatives: the answer is positive. −12 × −15 = 180

With more than two numbers, count the negatives: an even count gives a positive answer, an odd count gives a negative one.

Before you start, round each number to one non-zero digit and multiply:

An answer of 17,208 is close; 4,302 (a missing placeholder zero) is clearly wrong. For an exact check, divide the answer by one of the numbers: 17,208 ÷ 36 = 478. The Long Division Calculator shows that check worked out.

• Forgetting the placeholder zero in the second row (478 × 36 coming out as 4,302).
• Adding the carry before multiplying: 7 × 6 + 4 is 46, not 7 × 10 = 70.
• Carrying into the wrong row: clear or cross out old carries before starting the next row.
• Lining up decimal points as if you were adding. Count places instead.
• Misaligned columns when adding the rows. Squared paper helps.

Work these out on paper, then check each one with the Multiplication Calculator.

1. 64 × 7
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   448. 4 × 7 = 28 (write 8, carry 2), then 6 × 7 + 2 = 44.
2. 347 × 8
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   2,776
3. 52 × 34
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   1,768. Rows: 52 × 4 = 208 and 52 × 30 = 1,560.
4. 286 × 45
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   12,870. Rows: 1,430 and 11,440.
5. 1,024 × 203
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   207,872. Rows: 3,072 and 204,800; the tens row is all zeros.
6. 3.2 × 1.5
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   4.8. 32 × 15 = 480 with 2 places gives 4.80.
7. 0.25 × 0.4
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   0.1. 25 × 4 = 100 with 3 places gives 0.100.
8. −15 × 24
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   −360. One negative, so the answer is negative.