## Fractions

How do we use fractions? A fraction is the easiest way to express a part of a whole.

If you received 18/20 for your test, you got 18 marks correct out of the 20 marks available on the whole test.

Work through this tutorial to see how to add, subtract, multiply and divide fractions, and how to reduce your fraction to its simplest terms (i.e.using the smallest numbers)

- Simplifying fractions
- Adding fractions
- Subtracting fractions
- Multiplying fractions
- Dividing fractions

See also: Converting between fractions and decimals

### Simplifying fractions

Hi, I’m Martin Lindsay from the Study and Learning Centre at RMIT University. This is a short movie on simplifying fractions.

Let’s start by simplifying 27 divided by 36. The thing to notice about this fraction is that you can divide both top and bottom by nine, in other words, nine into 27 goes three and 36 divided by nine is 4. So we’ve reduced a more complicated fraction to a much simpler one, three quarters.

Let’s write this another way. 27 over 36 is divisible both top and bottom by nine. Nine into 27 goes three. Notice here we’ve cancelled the 27, in other words, we’re dividing by nine. And what we do to the top we do to the bottom, so we cancel the 36 and divide by nine, giving us four. Thus the answer is three quarters.

Here’s another example. We have to write 16 over five as a mixed number, or sometimes called a mixed fraction. 16 over five as you can see is a top heavy fraction, the number on the top is bigger than the number on the bottom. First of all what we need to do is divide the five into the 16, which gives us three and that gives us a remainder of one. And the remainder one is divided by the five, because the five is on the bottom of the fraction 16 over five. In other words, 16 over five is equal to three and one 5th.

Now let’s work in the reverse to what we were doing in the previous slide. Here we start with a mixed fraction or a mixed number and we need to turn it into an improper fraction. So we want to write two and four 7ths as an improper fraction. There’s our two and four 7ths and what we do is this. We multiply the seven by the two to give us 14. Once we’ve got the 14 we then add it to the four on the top of the fraction part of this number. So 14 plus four is 18. There’s the arithmetic on the right hand side. Two times seven in brackets, which we do first, then we add the four. That gives us fourteen plus four over seven, which gives us 18 over seven. In other words, the mixed fraction two and four 7ths is 18 over seven as an improper fraction.

Now try some questions for yourself. The answers to these questions are on the next slide. Thanks for watching this short movie.

### Adding fractions

Hi, I’m Martin Lindsay from the Study and Learning Centre at RMIT University. This is a short movie on adding fractions.

Let’s start by adding two basic fractions, a quarter plus two quarters. Notice in this case that the number on the bottom called the denominator is the same. We call that the common denominator, in other words, the number they both have in common is four. Once the denominator is the same then we can just go ahead and start adding the top numbers, called the numerators, in other words, one plus two gives us three over four. So we leave the denominator alone and we add the numerators. So the answer is three quarters.

But what happens if the denominators are not the same? Well, we need to now find the common denominator of five and four, meaning what number does five and four divide into. That number is 20, so the common denominator is 20. And this is how we go about doing it. Five goes into 20 four times, so we multiple five by four in the left hand fraction. If we multiple the bottom by four, we must multiply the top of the fraction by four, so two times four is on the top. With our three quarters fraction, we’re multiplying four by five to give us 20, therefore we multiply the top by five, three times five. In other words, we now have adding the eight and the 15 to give us the answer for our numerator, so eight plus 15 is 23, divided by 20 will give us one and three 20ths.

Let’s add whole number fractions or mixed fraction. One and one 3rd, plus two and three 5ths. The first job is to separate out the whole numbers from the fractions. So we now have one plus two, and the fractions are one 3rd plus three 5ths. So the whole numbers must be added up and the fractions must be added up. So we now have three, and now we go ahead and find the common denominator of three and five, which is 15. So as with the previous slide, we say three times five is 15, one times five is five. The second fraction is five times three is 15, and therefore we multiply the top of the fraction by three, three times three. This gives us three plus five over 15, plus nine over 15. And five over 15 plus nine over 15 gives us 14 over 15. So our answer is three and 14 15ths.

Now try some questions for yourself. The answers to these questions are on the next slide. Thanks for watching this short movie.

### Subtracting fractions

Hi, this is Martin Lindsay from the Study and Learning Centre at RMIT University. This is a short movie on subtracting fractions.

Let’s start with an easy fraction subtraction; two quarters minus one quarter. The thing to notice about this particular fraction subtraction is that the denominator for both fractions is in fact the same, which is four. If the denominator is the same then we can simply carry out the subtraction of the numerators for both fractions to take away one giving us an answer of one quarter, but just to reiterate in this particular case the fraction denominators are the same, so the subtraction for the numerators can be carried out straight away. However what if the two fractions have different denominators, in this case four and five are different denominators, so what we do first of all is look for a common denominator, that is the lowest number in which the four and the five will divide into, that number is 20, so we then carry out a little operation here. What I’ve done is to multiply the four by the five and in the second fraction I’ve multiplied the five by the four, so what I do to the bottom I must do to the top, so I’ve multiplied the three by the five for the first fraction and I’ve multiplied the two by the four for the second fraction. Now I have the same denominator on both, for both fractions, which is 20, because the denominator is the same I can then simply subtract the numerators, 15 minus eight.

What about whole number fraction subtractions, in this case two and three-fifths take away one and one-third. What we do first of all is to subtract the whole numbers, in this case it’s two take away one, then we go ahead and subtract the fractions, three‑fifths minus one-third. Here the common denominator is 15 and so with the first fraction, three over five, I’m multiplying that by three to make me 15, therefore I multiply the top by three. The second fraction, one-third, I multiply the bottom by five to give us 15, therefore I multiply the top by five, this gives me one plus nine‑fifteenths minus five-fifteenths, which is an answer of one and four-fifteenths.

Now try some questions for yourself. The answers to the questions are on the next slide. Thanks for watching this short movie.

### Multiplying fractions

Hi, I’m Martin Lindsay from the Study and Learning Centre at RMIT University. This is a short movie on multiplying fractions.

Let’s start by looking at two simple fractions, two 3rds times four 5ths. Notice with this fraction problem that there are no terms top and bottom that cancel. In other words, what we do is to multiply out the top line and multiply out the bottom line, in other words, the numerator which is the top line is multiplied out, two times four, and the denominator, the bottom line, is multiplied out, three times five. Again stress the point here that there are no terms that cancel. So this gives us four times two is eight, three times five is 15. So our answer to this problem is eight 15ths.

But what happens if we have terms that do cancel? Here we have the problem five over eight times two over five. And you should notice that there are terms that cancel here. The first one which is the obvious one are the fives top and bottom. So we go ahead with our cancelling. Five into five goes one, five into five goes one. So we now have one times two over eight times one. But notice again we have cancelling of the two and the eight. Two into two goes once, and two into eights goes four. So we’re cancelling more terms. That’s about as far as we can go, so we multiply out the top line, one times one is one, the bottom line is four times one is four. So the answer to this problem is a quarter.

How about multiplying out mixed numbers? Here we have mixed numbers one and three quarters multiplied by two and two 3rds. When we multiply mixed numbers we first of all turn them into improper fractions. So one and three quarters is seven over four, two and two 3rds is eight over three. So now we look to see whether any of these terms cancel. And as you can see four goes into four once, four goes into eight twice. Nothing else cancels, so we then multiply out the top line, seven times two is 14, the bottom line, one times three is three. But notice we give our answer as a mixed number, not an improper fraction, so 14 over three is four and two 3rds.

Now try some questions for yourself. The answers to these questions are on the next slide. Thanks for watching this short movie.

### Dividing fractions

Hi, I’m Martin Lindsay from the Study and Learning Centre at RMIT University. This is a short movie on dividing fractions.

Let’s start by dividing two simple fractions, four 3rds divided by two 5ths. We start by changing the division sign to a multiplication sign and then we take the reciprocal of the two 5ths, the last term. In other words, we flip the two over five upside, which becomes five over two. We then carry out the operation multiplying four over three by five over two, and as you can see, the twos cancel, two into two goes one, two into four goes two. Nothing else cancels, so we multiply out top and bottom, 10 over three and leave our answer as a mixed fraction. So the answer to this question is three and one 3rd.

Let’s divide two mixed fractions. Two and one quarter divided by one and one 5th. Before we actually carry out the division we must express our mixed fractions as improper fractions, so two and one quarter is nine over four divided by one and one 5th is six over five. Now we carry out our division by first of all changing the division sign to a multiplication sign and then taking the reciprocal of six over five, which becomes five over six. From there we look to see if anything cancels, and as you can see the nine and the six will divided by three, three into six goes two, three into nine goes three. Nothing else cancels, so we multiply out the top line, three times five is 15, the bottom line, four times two is eight. Thus an improper fraction which must be converted into a mixed fraction, which is one and seven 8ths. So the answer to this mixed fraction question is one and seven 8ths.

Occasionally you may see questions like this where there is a fraction divided by a whole number, and before you carry out the division problem you must express the whole number as a fraction. In other words, three over four divided by six, which is six over one. So you express the whole number as six over one. Then we carry out the operation by changing the division sign to a multiplication sign and taking the reciprocal of six over one, which is one over six. Look again for terms that cancel, here three goes into three once, three goes into six twice. Nothing else cancels, multiply out the numerator, one, the denominator, four times two is eight. So our answer to this question is one over eight.

Now carry out some questions for yourself. The answers to these questions are on the next slide. Thanks for watching this short movie.