IGCSE Maths Lesson: Fractions
1. Understand equivalent fractions and simplify with common factors
2. Understand mixed numbers
3. Identify common denominators
4. Use common denominators to add and subtract fractions:
5. Order fractions and calculate a given fraction of a quantity
6. Express a number as a fraction of another number
7. Convert a fraction to a decimal or percentage
8. Understand that multiplying by a fraction is the same as dividing by its inverse
9. Multiply and divide fractions and mixed numbers
Fractions are numbers represented as a number divided by another number. We are all happy with the idea of a half of something. If we have a pie and share it equally between 2 people they get half each We represent this numerically like this:
We refer to the number on the top as the Numerator and the number below as the Denominator.
Unlike with Integers we are going to start with multiplication and division of fractions.
But what if we want to find half of a half. In fractions, if we use the word 'of' then we are multiplying. Experience tells you that half of a half is a quarter, but how does this work mathematically:
We multiply the top line and the bottom line. However, there is another step as we should also display the fraction in the simplest possible terms. This is where factors and multiples from the previous lesson come in. ¼ cannot be simplified any more. But let’s try another sum:
15 over 30 is the equivalent of a half. For example, imagine a cake with 30 slices If you took 15 slices away , you've taken half of the cake. When answering questions, it is mathematical practice to display fractions in the simplest form. That is with the lowest denominator possible.
Try multiplying and simplifying the following:
Division of Fractions
So how do we divide a fraction by another fraction? The short answer is that you flip the second fraction upside down and multiply – easy isn't it!
This is the correct way to divide by a fraction. But why?
One way to think of it is to go back to the pie. If we share the pie out between 2 people, what are we doing mathematically? We are dividing by 2 or you could just as easily say multiplying by ½. So to divide by a half, we can flip the fraction and are therefore multiplying by 2.
Note: none of these examples can be simplified, however once the numerator exceeds the denominator the fraction can be expressed slightly differently, these are called mixed numbers. They are a combination of an integer and a fraction.
Adding and Subtracting Fractions
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It is easy to add up fractions with the same denominator. Simply add up the numerator and the denominator stays the same.
We know that a half plus a half = 1.
However, what if the denominators are different:
To perform this function, we need to convert the fractions into the same denominator. The easiest way to do this is to find the Common Multiple. For example, what is a multiple of 8 and 6? Well, 8 x 6 = 48
So, we convert the fractions into having 48 as the denominator. How do we do that? The rule to remember is that whatever we do to the denominator we must also do to the numerator.
We've changed how the fraction looks, but have only multiplied it by 1. We have not changed its value. Let us look at the other fraction:
Now we have both fractions with the same denominators, so we can add them up.
The same rules apply to subtraction.