IGCSE Maths Lesson: Fractions

Key Points:

1. Understand equivalent fractions and simplify with common factors

8 over 60 equals 2 over 15

2. Understand mixed numbers

7 over 5 equals 1 2 over 5

3. Identify common denominators

6 over 12 plus 6 over 18 equals 1 half plus 1 third equals 3 over 6 plus 2 over 6 equals 5 over 6

4. Use common denominators to add and subtract fractions: 

6 over 5 equals 5 over 5 plus 1 fifth equals 1 1 fifth
21 over 8 equals 8 over 8 plus 8 over 8 plus 5 over 8 equals 2 5 over 8

5. Order fractions and calculate a given fraction of a quantity

1 half greater than 1 fourth greater than 1 over 6
1 over 6 space o f space 18 space i s space 1 over 6 cross times 18 space equals space 18 over 6 equals 3 over 1 equals 3

6. Express a number as a fraction of another number

9 cross times 3 equals 27
9 space i s space 1 third space o f space 27

7. Convert a fraction to a decimal or percentage 

3 over 5 equals 0.6 equals 60 percent sign
1 fifth equals 0.2 equals 20 percent sign

8. Understand that multiplying by a fraction is the same as dividing by its inverse

cross times 1 half space i s space t h e space s a m e space a s space divided by space 2 over 1
divided by 1 half space i s space t h e space s a m e space a s space cross times 2 over 1
2 over 1 equals 2

9. Multiply and divide fractions and mixed numbers

18 over 48 cross times 40 over 48 equals 58 over 48
equals 48 over 48 plus 18 over 48 equals 1 9 over 24 equals 1 3 over 8

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: 

1 half


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: 

1 half cross times 1 half equals 1 fourth

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:  

3 over 5 cross times 5 over 6 equals 15 over 30 equals 1 half

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:

3 over 7 cross times 8 over 3
2 over 3 cross times 9 over 2
5 over 4 cross times 2 over 15

Answers:  

3 over 7 cross times 8 over 3 equals 24 over 21 equals 8 over 7
2 over 3 cross times 9 over 2 equals 18 over 6 equals 3 over 1 equals 3
5 over 4 cross times 2 over 15 equals 10 over 60 equals 1 over 6

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! 

1 third divided by 3 over 4 space equals space 1 third cross times 4 over 3 equals 4 over 9

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. 

cross times 1 half space i s space t h e space s a m e space a s space divided by 2 over 1
divided by 1 half space i s space t h e space s a m e space a s space cross times 2 over 1
2 over 1 equals space 2
 Try these: 

3 over 5 divided by 1 half equals
3 over 4 divided by 2 over 7 equals
1 over 6 divided by 3 over 5 equals

Answers:

3 over 5 cross times 2 over 1 equals 6 over 5
3 over 4 cross times 7 over 2 equals 21 over 8
1 over 6 cross times 5 over 3 equals 5 over 18

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. 

6 over 5 equals 5 over 5 plus 1 over 6 equals 1 1 over 6
21 over 8 equals 8 over 8 plus 8 over 8 plus 5 over 8 equals 2 5 over 8

Adding and Subtracting Fractions 

Check out this link

http://www.mathsisfun.com/fractions_addition.html

It is easy to add up fractions with the same denominator. Simply add up the numerator and the denominator stays the same.

1 half plus 1 half equals 2 over 2 equals 1

 We know that a half plus a half = 1.

However, what if the denominators are different:

3 over 8 plus 5 over 6 equals ?

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.

3 over 8 cross times 6 over 6 equals 18 over 48
D o n apostrophe t space f o r g e t space space 6 over 6 equals 1

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:

5 over 6 cross times 8 over 8 equals 40 over 48
D o n apostrophe t space f o r g e t space space 8 over 8 equals 1

Now we  have both fractions with the same denominators, so we can add them up.

18 over 48 plus 40 over 48 equals 58 over 48
equals 48 over 48 plus 18 over 48 equals 1 10 over 48 equals 1 5 over 24

The same rules apply to subtraction. 

Try these:

1 third plus 3 over 9 equals
2 over 5 minus 5 over 8 equals
1 over 7 plus 3 over 5 equals

Answers:

1 third plus 3 over 9 equals 1 third plus 1 third equals 2 over 3
2 over 5 minus 5 over 8 equals 16 over 40 minus 25 over 40 equals negative 9 over 40
1 over 7 plus 3 over 5 equals 5 over 35 plus 21 over 35 equals 26 over 35

 Video:

Last modified: Tuesday, 3 December 2019, 2:25 PM