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 more, 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 only thing to remember is to line up the decimal points in your two numbers before you start. That way, the decimal point in the answer will go immediately below the two in the sum.
Examples:
Example 1
123.5 + 234.2
As for any addition sum, 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 |
Example 2
234.8 + 147.9
234.8 + 147.9
H | T | U | . | t | ||
2 | 3 | 4 | . | 8 | ||
1 | 4 | 7 | . | 9 | + | |
Total | 3 | 8 | 2 | . | 7 |
Example 3
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 |
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 numbers work the same way as a number without a decimal. Simply multiply the numbers exactly as if there was no decimal point at all.
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. Dividing by 10 moves it one place to the left.
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’s a whole number. Multiply by 10 the number that you are dividing (the numerator) the same number of times. Then do the sum.
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 sum, 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 sum. 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 simply 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 sum.
For example, ^{1}/_{2} is actually 1 divided by 2, and it is also, of course, ^{5}/_{10} which is expressed as 0.5 in decimals.
So to convert a fraction to a decimal, you simply do the sum expressed in the fraction, 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.
The answer is therefore 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.
Continue to:
Fractions
Percentages