KS3. Number. 9. Adding & Subtracting Fractions

Let’s remind ourselves of what a fraction actually is.

It’s a way of expressing a part of a quantity, but keep in mind that this part can be bigger than 1.

So suppose we split a quantity (of chocolate, say) into 7 pieces.

Then if the number 1 means the total quantity of chocolate, then 1/7 means 1 of the 8 pieces we split it into.

And 3/8 means 3 of the 8th pieces we split it into.

So if we add 1/7 of the chocolate to 3/7 of the chocolate, how much chocolate will we have?

This is not the only way we use fractions, as we will see in the future, but it is perhaps the most important use of them.

Two other things we should know:

1.) When writing a fraction as the answer to a question, we should always write it in its simplest form. This means that if there are any common factors between the numerator and the denominator, we must first divide them out. So 3/60 will never be the final answer to a questions. What should it be? Let’s try writing some other fractions in simplest form.

2.) If we have numbers bigger than one written as a fraction, these can be written as mixed numbers or improper fractions. A final answer should always be given as a mixed number. To do addition and subtraction with these numbers, it may be easier to do the calculations separately for the whole number part and the fractional part, and then add the two together. Alternatively you could turn both numbers into improper fractions and then do the calculation. This will involve larger numbers, but less fiddling about at the end. The answer, though, should always be a mixed number.

Let’s try together changing some improper fractions into mixed numbers to make sure that we are happy with it.

Now let’s do our first exercise on addition and subtraction with fractions. It is exercise 7A on pages 102 and 103 of your textbook. Also I’ve copied it down below. If you finish the exercise there is an extension activity directly below.

Answers to above questions

Extension activity: Equivalent Fraction Puzzles

Answers to puzzles

Adding fractions with different denomiators

So, let’s suppose we wanted to add 3/8 and 7/12.

Again, let’s stop a moment to think about what this means. This time we will think about orange juice.

So imagine that you have two bottles of orange juice, both of the same size.

From one bottle you pour 3/8 (that is 3 “1/8 parts” of the orange juice into a bowl.

From the other bottle you pour 7/12 (that is 7 “1/12 parts” of the orange juice into a bowl.

Now how many bottles of orange juice are there now in the bowl?

Clearly, it’s a little bit difficult, because the size of the parts were different, so it’s not very useful just to add them together. In order to add parts together, the size of the parts must be the same.

So, what does this mean? It means instead of adding 1/8 parts and 1/12 parts, we need to change the way we write these fractions so the parts are the same.

And we are going to use the concept of lowest common multiple that we learned last term. What is the lowest common multiple of 8 and 12? Well these are the parts we want.

Now we just need to change the form of our fractions to put them into these parts. The main rule when changing the form of a fractions is that we can multiply and divide the numerator and the denominator of the fraction by any number (except for zero), but we must always multiply or divide both of them by the same number, that way we will only change the form, or the appearance of the fraction, but its size will stay the same.

Let’s try a few of these with the teacher:

Now you can have a go at exercise 7B on page 103 of your textbook, detailed below:

Anwers

Adding mixed numbers

Of course, we can also do this with mixed numbers or improper fractions, just remember to give your answer as a mixed number!:

Let’s try some:

Subtraction

Subtractions work exactly the same as addition, in that you need to find a common denominator and then you can subtract them. As before, let’s start without any mixed numbers. Here we have a mixture of additions and subtractions to practice with the teacher:

And let’s complete the exercise 7D on pages 104 and 105 of your textbook:

Answers:

Subtracting mixed numbers

And I am sure that if the above were no problem for you, then you will also have no problem subtracting mixed numbers. Let’s try some first with the teacher: