How to add decimals

When we add decimals, there's one thing to keep hold of: we keep the different parts separate. Wholes go with wholes, tenths with tenths, hundredths with hundredths. Once you do that, adding decimals is no harder than adding whole numbers, and the only extra thing to learn is what to do when a place overflows.

Keep the parts separate

When we work with decimals, it's important that we don't mix the parts up with each other. That means that we mustn't mix tenths with hundredths, and so on, because a tenth and a hundredth don't mean the same thing. So when we add decimals or subtract them, we keep the different parts separate. In practice that means writing the numbers so that wholes sit under wholes, tenths under tenths, and so on, and then each place is added to the same place in the other number. If the places are new to you, read decimal place value first.

Example 1: when nothing overflows

Let's add

$$1.234 + 2.542$$

In the first number, $1.234$, we have 1 whole, 2 tenths, 3 hundredths and 4 thousandths. We have to add that to $2.542$, which consists of 2 wholes, 5 tenths, 4 hundredths and 2 thousandths.

So we add each part to the matching part: $1 + 2 = 3$ wholes, $2 + 5 = 7$ tenths, $3 + 4 = 7$ hundredths and $4 + 2 = 6$ thousandths. That gives

$$1.234 + 2.542 = 3.776$$

Example 2: when a place overflows

There are also situations where there are more parts than there is actually room for on a place. For example

$$0.09 + 0.02 = 0.11$$

When there are more parts than there is room for, we simply move over to the next part. We can see that as soon as we add $2$ on top of $9$, we get too many parts on that place, so we have to move up to tenths. That is because $\tfrac{10}{100}$ is the same as $\tfrac{1}{10}$. So 10 hundredths is the same as 1 tenth:

$$\frac{10}{100} = \frac{1}{10}$$

Nine hundredths plus two hundredths is eleven hundredths. Ten of them become one tenth, and one hundredth is left over. So we get 1 tenth and 1 hundredth, which is $0.11$.

If you know equivalent fractions, you'll recognise this: $\tfrac{10}{100}$ simplified is $\tfrac{1}{10}$. Carrying from one place to the next is that same move.

Subtracting decimals

When we subtract decimals, we keep the parts separate in exactly the same way: wholes from wholes, tenths from tenths, and so on. The rule doesn't change, only the sign does.

Common mistakes

• "I can just add the digits and not worry about where the point is." The point tells you which digit is which part. Add wholes to wholes and tenths to tenths, so you can see which part is which.
• "$0.09 + 0.02$ gives $0.011$." No. Nine hundredths plus two hundredths is eleven hundredths, and ten hundredths is one tenth. So the answer is $0.11$, one tenth and one hundredth.

Related

What the places are worth is explained in decimal place value, and the overview sits in What are decimals?. The carry rule, ten hundredths make a tenth, is the same idea as equivalent fractions.