The same digits at a different size

Before this page: be able to multiply whole numbers with long multiplication, know what each digit is worth in integers and decimals, and know what happens to a number when it is multiplied or divided by ten.

Multiply 37 by 26 and you get 962. Multiply 3.7 by 2.6 and you get 9.62 — the same three digits, with a point dropped into them.

That is not a coincidence, and it is most of what you need. Decimals do not get a written method of their own. You run the whole-number method on the digits, then work out where the point belongs. The digits look after themselves; only the size is ever in doubt, and 962, 96.2, 9.62 and 0.962 are the same digits at four different sizes.

Choosing between them turns out to be a count, and most of this page is about where that count comes from. There is one surprise on the way: 0.3 × 0.4 = 0.12, which is smaller than either number it came from.

Making both numbers whole

Start by getting rid of the points. Multiply 3.7 by 10 and you have 37. Multiply 2.6 by 10 and you have 26. As on the powers of ten page it is the digits that move, one place to the left each time, while the point stays where it is between the ones and the tenths.

You have now made one number ten times too big and the other ten times too big as well, so whatever they multiply to will be a hundred times too big. Keep that hundred in view. Everything that happens at the end of the calculation is undoing it.

The ×100 going in and the ÷100 coming out cancel each other, which is why the answer can be trusted. On the way back down each row hands its own place back as the answer shrinks, so every line the board shows is one you could check for yourself. Every decimal multiplication has that shape; the only thing that changes is how big the power of ten is.

Counting the places

You will not want to write out the ×100 and the ÷100 every time, and you do not have to. Each decimal place in the question is one factor of ten standing between you and a whole number, and every ten you clear at the start is a ten you put back at the end. So the places the answer ends up with are simply the places the question had.

Count the decimal places in the question. Give the answer that many. The rule is only ever a count.

Do the counting before you multiply, not after. Counting afterwards means looking at the answer you have just written down and counting the places in that, which will agree with itself whatever you did.

Setting it out

When you write the multiplication down, the points are not there at all. That catches people out, because addition and subtraction in columns need the points lined up and this needs the opposite.

The difference is in what the columns are for. Adding tenths to tenths only works if the tenths sit above each other, so the points have to agree. In a multiplication every digit on top meets every digit underneath whatever column it is in, so there is nothing to line up: you write the two whole numbers the way you would write any two whole numbers, ones under ones.

Why the answer can be smaller

Here is the surprise. 0.3 × 0.4 = 0.12, and 0.12 is smaller than 0.3 and smaller than 0.4. Multiplying by a whole number bigger than 1 never does that to you.

It stops being strange as soon as you stop reading × as “makes bigger”. Multiplying by 0.4 means taking four tenths of something, and four tenths of anything is less than you had. Drawn as an area, you can count the answer rather than take it on trust.

Twelve squares out of a hundred. The 12 is nothing more than 3 × 4; the hundredths come from the way the square was cut, since a tenth of a tenth is a hundredth. That is the counting rule again, this time as a picture.

Whether a product comes out bigger or smaller than what you started with depends only on whether the multiplier is above or below 1, which has a page of its own.

When the answer ends in a zero

0.5 × 0.4 gives 5 × 4 = 20, and two places to count back. That lands on 0.20.

0.20 and 0.2 are the same number, so write the shorter one — but do it in that order. Tidy the 20 down to 2 first and you will count one place instead of two and finish at 2.0, which is ten times too big.

Two zeros turn up in that calculation doing opposite jobs. The one in front of the point is holding the ones column; take it away and the answer reads .2, which is easy to misread as 2. The one on the end is holding nothing, because there is nothing past it to keep in place.

Where it turns up

Decimals in a question are usually money or measurements, and both of them arrive with a unit that has to survive as far as the answer.

The count of places never depends on what is being measured. What the context decides is whether a trailing zero is kept or dropped, and that is a question about writing the answer down rather than about the arithmetic.

Multiply two chosen numbers

Use whole numbers or decimals. The calculation treats their digits as whole numbers first, then restores the combined number of decimal places in the product.

Common mistakes

The multiplication is the part that tends to go right. Nearly every lost mark here is the point landing in the wrong column, or a zero being treated as though it were doing a job it is not.

• Counting the answer instead of the question3.7 × 2.6 has two decimal places in it, so the answer gets two, however many digits the multiplication happens to throw up.
• Lining the points upThat is how you set out a column addition, and in a multiplication it puts the wrong digits above each other. Ignore the points until the answer.
• Losing a zero that holds a place0.2 × 0.3 is 6 with two places, which gives 0.06. Writing 0.6 is ten times too big: a zero has to go into the tenths column to push the 6 out into the hundredths.
• Tidying before counting0.5 × 0.4 gives 20, and both places are counted back before the zero on the end comes off. Counting one place because 20 “is really 2” gives 2.0 instead of 0.2.

Summary

1. A decimal multiplication is a whole-number multiplication with the point put back afterwards.
2. Making both numbers whole multiplies the calculation by a power of ten, and dividing the answer by the same power puts it back.
3. In practice that is a count: as many decimal places in the answer as in the question.
4. Multiplying by a number below 1 leaves less than you started with, because a tenth of a tenth is a hundredth.
5. Count the places first, then tidy any zero off the end — never the other way round.