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Maths help: How do you add fractions? How do you subtract fractions? Does the Numerator and Denomintor mean anything?

Updated on July 18, 2014

Lets start with - What is a fraction?

Fraction: means a numerical quantity that is not a whole number. It represents a small or tiny part, amount or proportion of something.

Another way to think of a fraction that links with your maths is a way of of dividing a whole into parts:

  • The bottom number or Denominator, represents how many equal parts you divide or cut that whole into.
  • The top number or numerator, represents how many of these parts you want.

What is the denominator and Numerator?
What is the denominator and Numerator? | Source

In fact when I teach this subject I get the students to read fractions as 'out of; .


For example:

  • One out of two = 1/2
  • Two out of three = 2/3
  • Three out of four = 3/4

This way you read it as it is. If you have a cake and cut it into four parts, then you can have one out of the four equal parts and your friends can have the other three parts out of the four cut.

How do I add fractions when the denominator is the same?

A general rule for adding or subtracting fractions is that the denominator MUST be the same number. If they are not then you cannot add them.

So when you are asked to add or subtract fractions when the denominator is already the same then it is a relatively easy ask.

What you must do is ...

  1. Add the top numbers and write the answer as the numerator of the answer.

  2. Keep the bottom number, or denominator, exactly the same.

Example 1:

 
 
 
 
 
1
+
2
=
3
5
 
5
 
5
Adding fractions
Adding fractions | Source

So in the above example, we worked out:

  1. 1+2=5
  2. We left the denominator alone as 5.
  3. And ended up with the fraction of 3/5

1/5 + 2/5 = 3/5

Example 2:

 
 
 
 
 
2
+
1
=
3
4
 
4
 
4
Adding fractions
Adding fractions | Source

So in the above example, we worked out:

  1. 2 + 1 = 3
  2. We left the denominator alone as 4.
  3. And ended up with the fraction of 3/4

2/4 + 1/4 = 3/4

It does get harder than this though ...

So remember rule 1:

The denominator MUST be the same number before you start working with fractions.

So what do we do if the denominators are different?

The steps you need to take here a little more complicated:

  1. Find the equivalent fraction for both that has the same denominator.
  2. Once they have the same denominator then you need to add the top numbers together (the numerator) .
  3. Leave the bottom numbers alone and use this in the answer (the denominator).

How do we do that first step?

To find out what equivalent fraction we need, so both fractions have the same denominator can be difficult. This is usually the step that a lot of people find really hard.

There are a number of ways we can have a look at this:

  1. Find the LCM (lowest common multiple) of the two denominators. (If you are not sure how to do this please take a look at the steps below)
  2. Multiply the the numerator and denominator by the denominator of the second fraction. Then multiply the numerator and denominator by the denominator of the first fraction. (This is an easy way to do the maths so I will start off by showing you what I mean with this example.) Once you have done this you will need to simplify the fraction though.


Multiplying the fraction by the denominator!

This will work best by showing you what I mean with this. And I will show you with a simple fraction to show you that it works:

Example 1:

 
 
 
 
 
 
 
 
 
 
 
1
+
1
=
4
+
2
=
6
=
3
2
 
4
 
8
 
8
 
8
 
4
 
 
 
 
 
 
 
 
 
 
 

So what did I do:

  • I multiplied the numerator and denominator of the first fraction (1/2) by the denominator of the second fraction (4)

1x4 = 4, 2x4 = 8, so we have 4/8.

  • Then I multiplied the numerator and denominator of the second fraction (1/4) by the denominator of the first fraction (2)

1x2 = 2, 4x2 = 8, so we have 2/8

That means we have 4/8 + 2/8.

And we can add these up because they both have the same denominator.

The next step is to add both numerators 4+2=6. And leave the denominator the same = 8. So we have the fraction of 6/8 when we add 1/2 and 3/4.

Of course we can simplify this fraction from 6/8 to 3/4 by dividing the numerator and denominator by 2.

Example 2:

 
 
 
 
 
 
 
 
 
3
+
1
=
9
+
5
=
14
5
 
3
 
15
 
15
 
15
 
 
 
 
 
 
 
 
 

So what did I do with this one:

  • First I multiplied the numerator and denominator of the first fraction (3/5) by the denominator of the second fraction (3)

3x3 = 9, 5x3 = 15, so we have 9/15

  • Next I multiplied the numerator and denominator of the second fraction (1/3) by the denominator of the first fraction (5)

1x5 = 5, 3x5 = 15, so we have 5/15

That means we have 9/15 + 5/15

And we can add these up because they both have the same denominator.

The next step is to add both numerators 9+5=14. And leave the denominators the same = 15. So we have the fraction of 14/15 when we add up 3/5 + 1/3.

And the good news with this is that we cannot simplify the fraction so job done.

Finding the LCM (Lowest common multiple)

So to help with working out the LCM I will include the same fractions as the examples above. That way you can compare both methods and use which ever one you find easier.

So how do you work out LCM of two numbers?

  • You take the two numbers you are working with and then write down a list of multiples after it.
  • If you write around 5-6 this usually is enough for most purposes, although you may find you might need more in your list.
  • Next you look at both lists and find the smallest number which is the same in both lists.
  • This is the LCM.

Example of working out LCM for the numbers 3 and 5:

3: 6,9,12,15,18

5:10,15,20,25,30

As you can see, 15 is the smallest common number in both lists. So 15 is the LCM.

LCM = least common denominator!
LCM = least common denominator! | Source

How does working out the LCM of two numbers help me with adding fractions.

Remember that you need to work out equivalent fractions in order to make sure the denominators on both fractions are the same. Well working out the LCM for the denominators will help you to do this.

Follow these steps to help you:

  1. Work out the LCM of both denominators first.
  2. Next, work out what how many times you had to multiply the denominator to get to to that number.
  3. Then multiply the numerator by the same value.
  4. One you have done this for both fractions you will be able to add them because both denominators will be the same.

(NOTE: there will be no need to simplify with this method so if that is what you find hard then this will be a good method for you)

Example 1:

1/2 + 1/4 =

LCM of 2 and 4 =

2: 4,6,8,10,12

4: 8,12,16,20,24

The LCM of both numbers is 4 because this is the lowest number that is the same on both lists.

I multiply 2 (the denominator of the first fraction) by 2 to get to 4, so I have to multiply 1 (the numerator of the same fraction) by 2 to get 2 in order to find the equivalent fraction. Ending up with 2/4.

As 4 is the first number in the list for the second denominator we do not need to change it as anything multiplied by 1 is the same.

So we need to add 2/4 + 1/4 = 3/4.

LCM
LCM | Source

Example 2:

3/5 + 1/3 =

5: 10,15,20,25,30

3: 6,9,12,15,18,21

The LCM of both numbers is 15 because this is the lowest number that is the same in both lists.

  1. I multiply 5 (the denominator of the first fraction) by 3 to get to 15, so I have to multiply 3 (the numerator of the same fraction) by 3 too. So I get 9. The equivalent fraction is 9/15
  2. I then have to multiply 3 (the denominator of the second fraction) by 5 to get to 15, so I have to multiply 1 (the numerator of the same fraction) by 5 too, to get 5. The equivalent fraction is therefore 5/15.

I can then add both fractions because both have the same denominator.

9/15+5/15 = 14/15.

You can work out LCM for any number of numbers!

LCM for three numbers
LCM for three numbers | Source

There has been a lot about adding fractions but nothing about subtracting!

Well when it comes to subtracting you just follow exactly the same steps:

  • Find the equivalent fractions, if the denominators are different, to make sure that they are the same.
  • They you subtract the numerators.
  • And leave the denominators alone.

Example 1

 
 
 
 
 
5
-
1
=
4
6
 
6
 
6
 
 
 
 
 
Subtracting fractions
Subtracting fractions | Source
  • Because both denominators are the same then you can skip to the second step here.
  • Take away the numerators, so 5-1=4
  • Leave the denominators alone, so keep 6.
  • Therefore the fraction we are left with is 4/6

5/6 - 1/6 = 4/6

Example 2

 
 
 
 
 
 
 
 
 
 
 
3
-
1
=
6
-
1
=
5
=
1
5
 
10
 
10
 
10
 
10
 
2
 
 
 
 
 
 
 
 
 
 
 
Subtracting fractions with a different denominator
Subtracting fractions with a different denominator | Source
  • We need to find the equivalent fractions so both denominators are the same first:

LCM of 5 and 10 is 10

5: 10,15,20,25,30

10: 20,30,40,50,60

  • So we have to multiply 5 (the denominator of the first fraction) by 2 to get 10, so we multiply 3 (the numerator of the same fraction) by 2 to get 6. That makes 6/10
  • We do not have to do anything to the second fraction as it is already a fraction of 10.
  • We end up with 6/10 - 1/10 which we can do.


  • We take away the numerators, 6-1 = 5
  • But leave the denominator alone, 10
  • This results in an answer of 5/10.
  • This can be simplified though down to 1/2.

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