How to multiply decimals
There is nothing new in the multiplying. You multiply exactly as you would whole numbers, and the decimal point is dealt with at the end, in one step.
Ignore the decimal points and multiply. Then count the digits after the point in both numbers together, and give the answer that many digits after its point.
How to do it
- Write the numbers one under the other, lined up on the right, not on the decimal point.
- Multiply as if the points were not there: 3.6 × 1.2 is worked as 36 × 12. A 0 in front is dropped too, so 0.7 is worked as 7.
- Count all the digits after the decimal point in both numbers. 3.6 has one and 1.2 has one, so two.
- Count that many digits from the right of the answer, and put the point there. If you run out of digits, write zeros in front until you can.
Worked examples
What is 2.4 × 3?
| 2. | 4 | |
| × | 3 | |
| 7. | 2 |
24 × 3 = 72. There is one digit after the point in 2.4 and none in 3, so the answer has one too. 2.4 × 3 = 7.2.
What is 3.6 × 1.2?
| 3. | 6 | ||
| × | 1. | 2 | |
| 7 | 2 | ||
| + | 3 | 6 | 0 |
| 4. | 3 | 2 |
36 × 12 = 432. One digit after the point in each number makes two, so count two digits in from the right. 3.6 × 1.2 = 4.32.
What is 0.4 × 0.3?
| 0. | 4 | ||
| × | 0. | 3 | |
| 0. | 1 | 2 |
4 × 3 = 12, and the answer needs two digits after the point. That takes both digits of 12, so a 0 goes in front of the point. 0.4 × 0.3 = 0.12.
What is 0.3 × 0.2?
| 0. | 3 | ||
| × | 0. | 2 | |
| 0. | 0 | 6 |
3 × 2 = 6, but the answer needs two digits after the point and 6 is only one. Write zeros in front until there are enough, and one more before the point. 0.3 × 0.2 = 0.06, not 0.6.
Why counting the places works
3.6 is 36 tenths, and 1.2 is 12 tenths. Tenths times tenths are hundredths, because a tenth of a tenth is a hundredth. So 3.6 × 1.2 is 36 × 12 = 432 hundredths, which is 4.32.
Every digit after a point is one more ÷ 10. Multiplying the whole numbers leaves all of those divisions still to do, one for each place in either number, and putting the point that many digits in from the right does them.
Check by rounding: 3.6 × 1.2 is a bit more than 3.6 × 1 and a bit less than 4 × 1.2 = 4.8, so 4.32 fits, and 43.2 and 0.432 do not.
Where mistakes happen
- Lining the points up. That is right for adding decimals and wrong here. Line the numbers up on the right, as for any long multiplication.
- Counting the places in only one number. 3.6 × 1.2 needs two, one from each.
- Taking zeros off before placing the point. 2.5 × 4 is 25 × 4 = 100, with one place: 10.0, which is 10. Drop the zeros first and you get 1.0.
- Expecting the answer to be bigger. Multiplying by a number less than 1 makes it smaller, so 0.4 × 0.3 = 0.12 is smaller than both.
Questions people ask
Do I need to line up the decimal points?
No. Line the numbers up on the right and leave the points where they fall. Only adding and subtracting line the points up.
What do I do with a 0 before the point, like 0.7?
Leave it out of the multiplying, so 0.7 is worked as 7, but still count the 7 as a digit after the point.
Why did my answer come out smaller than both numbers?
Because both were less than 1. Taking 0.3 of 0.4 is taking less than all of it, so the answer has to be less than 0.4.