What are Fractions?
Fractions are similar to decimals in that they are used to show smaller parts of a whole number.
The decimal 0.2 has 0 units and 2 tenths. This can be written as a fraction like this:
The top number is called the numerator and the bottom number is called the denominator just to make things complicated.
But to help us understand fractions look at it as the top number being ‘how many’, the line in the middle as being ‘out of’, and the bottom number being ‘the total’.
The above fraction is 2 tenths. It means 2 parts out of a total of 10.
Here are some fractions:
This is one tenth (one part out of 10)
This is one half (1 part out of 2)
This is one third (one part out of 3)
This is one quarter (one part out of 4)
This is one fifth (one part out of 5)
This is two fifths (2 parts out of five)
Fractions bigger than 1 either have a bigger top number than bottom number:
Or we would normally write the unit first and then the fraction:
Both mean 2 and 2 tenths.
Finding fractions of numbers
To find a fraction of a number you can follow this method:
- Divide the number by the denominator in the fraction (the bottom number).
- Times the number by the numerator in the fraction (the top number).
Question: What is ¾ of 12?
To find three-quarters of 12 we are being asked to split 12 in to 4 equal parts and then to find the value of three of those parts.
To split a number in to 4 equal parts just means we need to divide it by 4:
12 ÷ 4 = 3
Then to find the value of three lots of that part we simply times the result by 3:
3 x 3 = 9
So three-quarter of 12 = 9
Let’s look at this visually.
We have 12 alien eggs:
We want ¾ of them so we split them in to 4 equal parts:
Then we want to know how many eggs are in three 3 lots of 1 part so we times the value of 1 part (3 eggs) by 3.
3 x 3 = 9
Equivalent means the same.
Equivalent fractions look different but they have the same value.
If you think about it, 1/2 actually has the same value as 2/4
If we have a cake made up of 2 slices and we eat one, we have eaten 1/2:
If we have a cake made up of 4 slices and we eat 2 we have still eaten a half of the cake, exactly the same amount:
So we can say that and are equivalent fractions.
To find out if fractions are equivalent we must change them if needed so that they use the same denominator
To change fractions we either multiply both the top and the bottom by the same number or we divide the top and the bottom by the same number (dividing fractions like this is called simplifying them).
Remember, what we do to the top we must do to the bottom.
With and we can do this by simplifying so that it is in the form ‘something’ over 2.
To get to be ‘something’ over 2 we are dividing the denominator 4 (the bottom number) by 2… so we must divide the numerator 2 (the top number) by 2 also.
2 ÷ 2 = 1.
So can be written as … they are equivalent fractions.
Simplifying fractions means writing them in their simplest form.
We have already seen this when we looked at equivalent fractions, we simplified to .
Let’s work through a question involving simplifying.
Question: Write in its simplest form.
We need to look at the denominator (the bottom number) and the numerator (the top number) and say what number can we divide both top and bottom by? When both numbers are even numbers like they are here it is a good idea to start with 2:
We are not finished yet though because the fraction is not yet in its simplest form – both top and bottom numbers can again be divided by 2:
Now there is no number that will divide in to both 5 and 2 to further simplify the equation, so this is the equation in its simplest form.
Comparing and Ordering Fractions
When asked to compare the size of fractions you must make the denominator (the number on the bottom) the same for each fraction.
You can’t compare fractions where the bottom number is different.
Let’s look at an example question.
Question: Put , and in order of size.
We first need to find a way to change the fractions so that they have the same denominator, then we can compare their sizes. This is similar to simplifying but in this case we can divide or multiply the number on the top and bottom of the fraction by the same number to get a new fraction.
With this question I am going to leave as it is. This is because it cannot be simplified any further and I can see that the other two fractions can be changed in to tenths.
Next we need to change into .
To change to , we are multiplying the bottom denominator by 2:
Remember, what we do to the top we must do to the bottom:
Now we also need to change in to .
To do this we divide the the top and bottom number by 2:
Now we have the fractions written in the same form so we can order them by the size of the numerator (the top number):
is the smallest.
is in the middle ().
is the biggest ().
Adding and Subtracting Fractions
To add or subtract fractions they must be of the same type.
You can’t add fractions with different deniminators (bottom numbers). They must be changed first so that they have the same bottom number.
Let’s start with an easy example question to work through:
Question: What is + ?
Well as they have the same deniminator, we just add the top numbers and the bottom number stays the same:
Now we are going to work through a more complicated question where the fractions are in different forms.
+ = ? (write you answer in the simplest form)
First we need to change the fractions to the same type. In this case it is a bit trickier to do that than the ones we have looked at before.
But we can change to .
We just need to work out how the denominator 12 can be changed to 8.
12 ÷ 3 = 4
4 x 2 = 8
So if we divide 12 by 3, and then times it by 2 it equals 8.
Now we can apply the same steps to the top number to change the fraction to eighths:
9 ÷ 3 = 3
3 x 2 = 6
So is equivalent (the same as)
Our sum is now a lot easier, we can just add the numerators:
When the numerator is bigger than the denominator, as it is here, we know we have a value greater than 1.
Do you understand why? is one whole. In the fraction we have 1 whole and .
We were asked to write our answer in the simplest form, well can be simplified, both top and bottom numbers can be divided by 4 to give . So our final answer is:
The above example question uses bits of knowledge from several sections in this lesson on fractions. If you didn’t understand any of the parts do go back and read about them again.
Subtracting fractions is the same as adding except that we minus the smaller numerator from the bigger one once we have got the fractions in the same form.