Multiplying and Dividing Decimals

Decimals turn up everywhere the moment you start scaling things. Six bottles of juice at £1.50 each — how much? A £4.80 bill split between friends — how much each? Three tenths of a four-tenths bar of chocolate — how much chocolate? Every one of those is a decimal being multiplied or divided.

It sounds like it might be fiddly, but there is a neat trick hiding in the place value that puts the decimal point in exactly the right spot — no guessing. Let's meet it with the little sum 0.3 \times 0.4.

Multiplying: ignore the points, then count them back

To multiply two decimals, forget the decimal points for a moment and multiply the numbers as if they were whole numbers. Then count how many decimal places the two original numbers had together, and put that many decimal places back into the answer.

So for 0.3 \times 0.4: ignore the points and multiply 3 \times 4 = 12. Each number had one decimal place — two in total — so the answer gets two decimal places:

0.3 \times 0.4 = 0.12

Multiplying or dividing by 10, 100 or 1000 is even simpler: it just shifts the digits past the point. Multiplying by 10 moves the point one place to the right; dividing by 10 moves it one place to the left:

2.5 \times 10 = 25 \qquad 2.5 \div 10 = 0.25

Each extra zero in the power of ten shifts the point one more place (\times 100 two places right, \div 1000 three places left).

Worked example — a price times a quantity

Six bottles of juice at £1.50 each: 1.5 \times 6. Ignore the point and multiply the whole numbers: 15 \times 6 = 90. Only one number, 1.5, had a decimal place — one place in total — so put one place back:

1.5 \times 6 = 9.0 = 9

The six bottles cost £9. (A whole number of pounds — how tidy.)

Dividing by a decimal: make the divisor whole first

To divide by a decimal, first turn the divisor (the number you are dividing by) into a whole number. You do this by multiplying both numbers by 10, 100 or whatever it takes. Scaling both up by the same amount keeps the answer exactly the same — like enlarging a photo, the shape doesn't change.

For example, 4.8 \div 0.6. Multiply both numbers by 10, so the divisor 0.6 becomes the whole number 6 and 4.8 becomes 48. Now it is an easy whole-number division:

4.8 \div 0.6 = 48 \div 6 = 8

Trap 1 — count the decimal places, all of them.

The answer to a multiplication gets the total number of decimal places from both numbers. In 0.3 \times 0.4 that is 1 + 1 = 2 places, so the answer is 0.12 — not 1.2 (that forgets a place) and not rounded down to 0.1 (that throws a place away). Count carefully and keep every place.

Trap 2 — multiplying can make things SMALLER.

We're taught that "times" makes numbers bigger, so 0.3 \times 0.4 = 0.12 looks wrong — the answer is tinier than both numbers you started with! But it's correct. When you multiply two numbers that are each less than 1, the answer is always smaller than both. Think of it in words: you're taking "three tenths of four tenths" — a part of a part is a smaller part. Multiplying only grows a number when you multiply by something bigger than 1.

There's a lovely reason the rule is true, not just a trick to memorise. Each decimal place means "divide by 10". So a tenths number is (\text{something}) \div 10, and another tenths number is (\text{something}) \div 10 too. Multiply them and the tens underneath multiply as well:

\frac{3}{10} \times \frac{4}{10} = \frac{3 \times 4}{10 \times 10} = \frac{12}{100} = 0.12

Tenths times tenths gives hundredths — two zeros underneath, two decimal places. That is the whole rule, straight from place value.

This same powers-of-ten magic is exactly why the metric system is so beautifully easy: metres, centimetres and millimetres all step by tens, so converting between them is just shifting the decimal point — no awkward multiplying by 12 or 16 like the old units. Powers of ten make everything slide.

See it explained