CONCEPT OF A FRACTION
If you cut a cake into two equal pieces and eat one of them, you have eaten 1/2 (half) a cake.
If a cake is cut into five equal pieces and you eat three of them, you have eaten 3/5 (three fifths) of a cake.
1/2 and 3/5 are examples of fractions - parts of a whole.
HOW TO READ FRACTIONS
To read a fraction in English you have to read the numerator and then the denominator with the ordinal. You can use plural if you have more than one part. For example, 1/3 is one third but 2/3 is two thirds. When the denominator is 2 you have to read “half”, 1/2 is one half, 5/2 is five halves. If the fractions is very big you can use “over”, 4/11 is four over eleven.
Equivalent fractions
Cutting the cake into six equal pieces and eating two is equivalent to cutting the cake into three equal pieces and eating one. You eat the same amount of cake in both cases.
Question
If the cake is cut into 12 equal pieces, how many will we have to eat in order to have the equivalent of 1/3 of the cake?
Common factors and simplest form
Common factors
The factors of a number are those numbers that divide into it exactly.
Numbers have common factors if the same number divides into both of them.
So 4 is a common factor of both 8 and 12, as it divides into both of them. 2 is a common factor of both 2 and 6, as it divides into both of them.
Simplest form
You know that 4/12 = 2/6 = 1/3
4 and 12 have a common factor (4), so 4/12 can be written as 1/3 (divide the top and the bottom by 4).
2 and 6 have a common factor (2), so 2/6 can be written as 1/3 (divide the top and the bottom by 2).
However, 1 and 3 have no common factors, so 1/3 cannot be simplified. When a fraction cannot be simplified we say that it is its simplest form.
Mixed numbers and improper fractions
A whole number can be written as 2/2, 3/3, 4/4, etc.
So 1 2/3 can be written as
3/3 + 2/3 = 5/3
Mixed numbers
1 2/3 is known as a mixed number, because it is made up of a whole number and a fraction.
Improper fractions
5/3 is called an improper fraction, because the top number is bigger than the bottom number.
Converting from a mixed number to an improper fraction
You can write the whole number part as a fraction, then add the fractions together.
1 2/3 = 3/3 + 2/3 = 5/3
Here is another example:
2 1/4 = 1 + 1 + 1/4 = 4/4 + 4/4 + 1/4 = 9/4
Converting from improper fractions to mixed numbers
You can separate out the fraction into smaller fractions, like this:
17/5= 5/5 + 5/5 + 5/5 + 2/5 = 3 2/5
Another way to convert an improper fraction is to find how many whole numbers you get, by using a division.
For example let’s convert 17/5 to a mixed number again.
We start by dividing the top number by the bottom number.
17 divided by 5 is 3 remainder 2.
So the whole number part is 3, and the remainder 2 means there are 2/5 left over.
So the answer is 17/5 = 3 2/5
Question
Write 20/7 as a mixed number.
Ordering fractions
Which fraction is bigger, 3/4 or 5/7 ?
It is hard to answer this question just by looking at the fractions. However, if you write the fractions with the same bottom number, the question will be easy.
3/4 has a denominator of 4, and 5/7 has a denominator of 7.
4 and 7 both divide into 28, so rewrite the fractions with a denominator of 28.
3/4= 21/28
5/7= 20/28
It is easy to see that 21/28 is bigger than 20/28.
Therefore 3/4 is bigger than 5/7.
To compare fractions, first write them with the same number at the bottom.
Adding and subtracting
It is hard to picture what the answer is if you add 1/2 and 1/3. Rewriting the fractions with a common bottom number (in this case, 6) makes it easier to see the answer.
Remember: You can only add and subtract fractions when the bottom numbers are the same.
So to add or subtract fractions:
- Change the fractions so they have the same bottom number.
- Add or subtract the top numbers.
Example
1/2 + 1/3 = 3/6 + 2/6 = 5/6
7/10 - 2/5 = 7/10 - 4/10 = 3/10
Question
What is 1/4 + 1/3 = ?
Multiplying fractions
1/2 of 1/2 = 1/2 × 1/2 = 1/4
2/3 of 4/5 = 2/3 × 4/5 = 8/15
Multiply the top and bottom numbers then simplify where necessary.
Question
Calculate 3/4 × 2/5 = ?
Dividing fractions
When you divide 10 by 2, you are working out how many 2’s there are in 10.
10 ÷ 2 = 5, so there are five 2’s in 10.
In a similar way, when dividing 2 by 1/2, you are working out how many 1/2’s there are in 2.
There are four 1/2’s in 2, so 2 ÷ 1/2 = 4.
If you divide 1 1/2 by 1/4 you are working out how many 1/4’s there are in 1 1/2 .
There are six 1/4’s in 1 1/2, so 11/2 ÷ 1/4= 6.
Do you see a pattern? Let’s write out those calculations a different way.
- 2 ÷ 1/2 = 4 so 2 ÷ 1/2 is the same as 2 × 2
- 11/2 ÷ 1/4 = 3/2 ÷ 1/4 = 6
so 3/2 ÷ 1/4 is the same as 3/2 × 4 = 12/2 = 6
Remember: To divide fractions, turn the second fraction upside down, then multiply.
Question
Calculate 3/4 ÷ 4/5
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