Addition and Subtraction of Fractions:

To go back to our above example,.let's say that you have eaten one slice of the pizza and your friend Tony has eaten two slices. Together you have eaten three slices, Or to express it in terms of fractions;

Together, you have eaten three eighths of the pizza. How much is left? Well, there were eight slices to begin with and if, between the two of you, you have eaten three, that means:

and there are five slices left. In terms of fractions:

and there are five eighths of the pizza left.

This was easy to calculate, but things become a little more difficult, when the denominators of the two fractions to be added or subtracted are not the same. Let us show you what has to be done:

Let us this time assume that your pizza has been delivered, but has not been sliced. You are feeling very hungry and eat one third of the pizza. Your friend Tony is even more starved and he eats one half of the pizza. How much of the pizza have you consumed and how much is left?

The two fractions to be added:

cannot be added in the way we have added the 1/8 and 2/8. In math, you always have to be careful to add or subtract the SAME UNITS! Because 1/2 and 1/3 have different denominators, they are not expressed in the same units and they first have to be "translated". How? Well, we have seen that we can add fractions, if the denominators are the same. Can we somehow change or manipulate the two fractions, so that they would end up having the same denominator?

If we multiply the denominator of the first fraction (2) by 3, we will get the result of 6.

If we multiply the denominator of the second fraction (3) by 2, we get the same result of 6.

If we perform these two calculations, we will then end up with two fractions with the same denominator of 6 and they can then be added:

Let's go ahead and do this. But remember, if you multiply the denominator by a number, you must multiply the numerator of that same fraction by the same number! That means we can have to multiply both the numerator and denominator of the first fraction by 3 and we will therefore end up with 3/6.

We will then multiply both the numerator and the denominator of the second fraction by 2 and get 2/6.

We now have two fractions, 3/6 and 2/6 that we can add, because they have the same denominator:

That means that you and Tony have eaten 5/6 of the pizza and left how much?

Well, the entire pizza represents the whole and is equal to 1. How could we deduct 5/6 from 1? Again, we would have to express 1 in terms of sixths. How many sixths is 1? Obviously 1 = 6/6. (If we are a group of six people and order one pizza and want to make sure that each person has the same quantity of the pizza to eat, how many slices would we divide the pizza into? 6 And what would each slice represent? One sixth. Therefore one entire pizza is 6/6).

We therefore subtract:

and we can see the 1/6 of the pizza is left.

You can now see that subtraction of fractions works in exactly the same way as the addition. It is essential to find a common denominator for the two fractions!

To find a common denominator, it is necessary to find a common multiple of both denominators. Let us say that you have the following example:

The common multiple of 3 and 2 is 6, therefore in order to have both fractions expressed with the same denominator of 6, we have to express both as their equivalents in sixths. To do that:

For 1/3, we see that we have to multiply the denominator of 3 by 2 (to get a denominator of 6) and therefore we have to multiply the numerator by 2 as well, to get:

For 1/2, we see that we have to multiply the denominator of 2 by 3 (to get a denominator of 6) and therefore we have to multiply the numerator by 3 as well, to get:

Now we can add:

Look at the next example:

To find the common multiple of 3 and 5, let us first look at the multiples of 3 and see which is the first one divisible by 5:

And we see that 15 is the first multiple of 3, which is divisible by 5 and is therefore their common multiple. Sometimes, we have to multiply the two common denominators together, to find their common multiple. Now let us finish this example:

To express 1/3 as a fraction with a denominator of 15, we have to multiply both the numerator (1) and denominator (3) by 5:

To express 2/5 as a fraction with a denominator of 15, we have to multiply both the numerator (2) and denominator (5) by 3:

Now we have both fractions expressed with the same denominator of 15 and we can add them:

Now try the following examples:

Examples:

Answers: