Decimals
See also: FractionsFractions and decimals are two different ways to represent parts of a whole number. Decimals are a way to express tenths, hundredths, thousandths (and beyond) of a unit.
Working with decimals may look a bit complex to start with but, don’t worry, they’re only numbers and they obey rules like other numbers.
Working with Decimals
Adding and Subtracting Decimals
Decimals extend the number system beyond the simple ‘hundreds, tens, units’ into ‘tenths of units’, ‘hundredths of units’ and so on.
Working with decimals is therefore essentially the same as working with any other number.
After looking at our pages on Numbers, Addition and Subtraction, you would have no concerns about adding thousands to the mix, so why worry about tenths and hundredths?
If you were adding numbers without decimals, you would start with the units, and move along to tens, then thousands and so on. The same rule applies if there are decimals. Add them first, then units, then tens and so on.
The most important rule to remember is to line up the decimal points in your calculation, ensuring that the decimal point in the answer also lines up with the decimal points above it.
Example 1 - Straightforward addition
123.5 + 234.2
As for any addition calculation, align the numbers and add the columns starting from the right.
Hundreds | Tens | Units | Point | tenths | ||
1 | 2 | 3 | . | 5 | ||
2 | 3 | 4 | . | 2 | + | |
Total | 3 | 5 | 7 | . | 7 |
123.5 + 234.2 = 357.7
Example 2 - Addition with different decimal places
234.8 + 147.96
In this example, we are adding a number that has one decimal place to a number that has two decimal places. Remember, it doesn’t matter how many decimal places we are dealing with, or whether the numbers involved have a different amount of decimal places. The most important part of the calculation is to line up the decimal points. If it helps you to line up the columns, you can write a zero in the hundredths column of the first number, or you can leave that box empty.
H | T | U | . | t | h | ||
2 | 3 | 4 | . | 8 | 0 | ||
1 | 4 | 7 | . | 9 | 6 | + | |
Total | 3 | 8 | 2 | . | 7 | 6 |
234.8 + 147.96 = 382.76
Example 3 - Subtraction
72.347 − 64.012
Subtract in the same way as with whole numbers, but make sure the decimal place is in the right place.
T | U | . | t | h | th | ||
7 | 2 | . | 3 | 4 | 7 | ||
6 | 4 | . | 0 | 1 | 2 | - | |
Total | 0 | 8 | . | 3 | 3 | 5 |
72.347 − 64.012 = 8.335
If you're confused about 'carrying over' when adding or subtracting see our pages Addition and Subtraction for help.
Multiplying Decimals
When multiplying and dividing decimals, the calculation works in the same way as with whole numbers. We multiply the numbers as if there was no decimal point at all. At the end of the calculation, we make sure that we have the decimal point in the correct place in our answer:
Starting with the answer that you have obtained by multiplying the numbers, move the decimal point the same number of places to the left as there are numbers after the decimal point in the two factors.
Example 1
0.5 x 0.5
5 x 5 is 25. There are two numbers after the decimal point, one in each of the multiplying numbers, so move the decimal point two places to the left, from 25, and the answer is 0.25
Example 2
1.2 x 0.25
First remove the decimal points 12 x 25 = 300
This time, there are three digits after the decimal place in the multiplying numbers, one in 1.2 and two in 0.25.
The decimal point in 300 is after the second zero, making it 300.0
Move the decimal point three places to the left, and the answer is 0.3
Dividing Decimals
Multiplying and dividing by 10
Multiplying by 10 moves the decimal point one place to the right (increasing the original number by a factor of 10). Dividing by 10 moves it one place to the left (decreasing the original number by a factor of 10).
You can use this fact to make dividing decimals a whole lot easier. Multiply by 10 the number that you are dividing by (the denominator) until it is a whole number. Multiply by 10 the number that you are dividing (the numerator) the same number of times. Then do the calculation.
Example:
50.22 ÷ 0.2
If you’re using the standard format for division, (see our page on division) where your answer goes above a line over the number you’re dividing, then the decimal point goes exactly above the one in the number you’re dividing:
T | U | . | t | h | |
0.2 | 5 | 0 | . | 2 | 2 |
You can simplify this calculation if you multiply 0.2 by 10 once to make 2. You therefore multiply 50.22 by 10 as well, to get 502.2
H | T | U | . | t | |
2 | 5 | 1 | . | 1 | |
2 | 5 | 0 | 2 | . | 2 |
Then do the calculation. It is much easier to divide by 2 than 0.2.
The answer is: 251.1
Top Tip
If you’ve done a multiplication or division involving decimals, then check to see if the answer looks about right. In other words, if you took away the numbers after the decimal point, and rounded up or down to a whole number, would it still be about right?
If your answer looks much too big or too small, then check the position of your decimal point. It may well be a position out in either direction.
Converting Between Fractions and Decimals
Converting from decimals to fractions is fairly straightforward. Any number can be expressed as a fraction by simply putting it over one.
For example:
2 = ^{2}/_{1}
21 = ^{21}/_{1}
The same rule applies to decimals.
Put the decimal over one, and then multiply both top and bottom by 10 until you no longer have a decimal point. Then, if possible, convert your fraction to a mixed number and/or reduce it down to its smallest form.
For example:
0.25 = ^{0.25}/_{1} = ^{2.5}/_{10} = ^{25}/_{100} = ^{1}/_{4}
1.25 = ^{1.25}/_{1} = ^{12.5}/_{10} = ^{125}/_{100} = ^{5}/_{4} = 1^{1}/_{4}
See our page on Fractions for more.
Converting from Fractions to Decimals
Converting from fractions to decimals is slightly harder, but gets easier once you realise that a fraction is actually a division calculation.
For example one half,^{1}/_{2}, is actually 1 divided by 2, which is also the same as ^{5}/_{10}, or five tenths, which is expressed as 0.5 in decimals This is because decimals are based on multiples of ten. (See our pages on An Introduction to Numbers and Systems of Measurement for more information.)
So to convert a fraction to a decimal, consider the fraction as a division calculation, adding zeros after the decimal point if necessary to complete it.
Example 1
^{2}/_{5} = 2.0 ÷ 5
5 goes into 20 four times, and the decimal point goes in the same place in the top line.
The answer is therefore 0.4
Example 2
^{4}/_{25} = 4.00 ÷ 25
25 goes into 40 once, leaving 15 as a remainder.
25 goes into 150 six times exactly. Finally, check the position of the decimal point is correct.
The answer is therefore 0.16
There is always more than one way!
As we practise calculations like this more and more, we start to spot ways of making it easier to work out the answer. Considering the example above, instead of doing the calculation step by step in the conventional way, we can stop and think “is there another way I can easily find out how many times 25 goes into 400?” We can put our mental arithmetic skills to work: With practise we will remember that there are 4 lots of 25 in 100, because 25% is another way of writing ¼. If there are four 25s in 100, then there must be 4 × 4 lots of 25 in 400, i.e. 16. Moving the decimal place two places to the left gives us 0.16
If the division is troubling you, take a look at our page on Division for a quick reminder.
Points to remember:
- Decimals express tenths, hundredths, thousandths and so on of units.
- Treat them as any whole number, but watch the position of the decimal point in your answer.
- If the answer looks wrong, check the position of the decimal point.