# Fractions

See also: DecimalsLike decimals, fractions describe parts of a whole.

Our page Numbers an Introduction explains that they are expressed as a division sum, one number divided by another, or commonly expressed as one number over another.

A half, for example, is written as ½. One divided by two, or often said as 'one over two'.

Fractions, like decimals, are only numbers. They conform to rules. Although the rules may seem slightly more complicated for fractions, with a little practice they are relatively easy to grasp.

### Some Basic Terms and Rules of Fractions

- The numbers in a fraction are called the
*numerator*, on the top, and the*denominator*, on the bottom.^{numerator}/_{denominator} *Proper fractions*have a numerator*smaller*than the denominator.

Examples include^{1}/_{2},^{3}/_{4}and^{7}/_{8}.*Improper fractions*have a numerator*larger*than the denominator.

Examples include^{5}/_{4},^{3}/_{2}and^{101}/_{7}.

Improper fractions can always be expressed as a whole number together with a proper fraction - and usually you should do this.

In our example:

^{5}/_{4} is the same as 1^{1}/_{4}

^{3}/_{2} = 1^{1}/_{2}

^{101}/_{7} = 14^{3}/_{7}

- When working with fractions, they are always expressed as the
*smallest possible set of numbers*. In other words, if the bottom number divides by the top number, divide it down (*reduce it*) until you can no longer do so.

Example:

^{2}/_{14}=^{1}/_{7}. The numerator (2) and denominator (14) are both divided by 2.

In the same way:^{2}/_{8}=^{1}/_{4}

^{3}/_{24}=^{1}/_{8}. Here both numerator and denominator are divided by 3.

- Sometimes the bottom number does not divide by the top number, but they both divide by some other number. In mathematical terms, this means that they have a
*common factor*.

In such cases, divide both numbers by the common factor until one or both are either prime numbers, or they have no more common factors

^{24}/_{60}=^{12}/_{30}=^{2}/_{5}. Divide first by 2 and then by 6.

^{21}/_{35}=^{3}/_{5}. Divide by 7.

^{21}/_{31}. Cannot be reduced, as 31 is a prime number, so cannot be divided by anything except itself and one.

^{16}/_{33}. Although both numbers have factors, they have no common factor, so this fraction cannot be reduced.

## Adding and Subtracting Fractions

See our pages, Addition and Subtraction for more general help.

The easiest fractions to add or subtract are those with the same denominator. You simply add or subtract the two numerators, and place them over the same denominator.

For example:

^{3}/_{8} + ^{2}/_{8} = ^{5}/_{8}

Likewise, the same applies when subtracting fractions

^{7}/_{8} – ^{5}/_{8} = ^{2}/_{8}. This can be simplified further to ^{1}/_{4}

However, it’s a bit more of a challenge when the two numbers don’t share a common denominator.

In such cases, you need to find the **lowest common denominator**, or LCD. That is, the smallest number which divides by both denominators.

This may be straightforward; for example, if you are adding ^{1}/_{4} and ^{1}/_{2}, then 4 divides by 2, and the lowest common denominator is therefore 4. So ^{1}/_{4} + ^{2}/_{4} = ^{3}/_{4}.

Sometimes it is not so easy to spot the lowest common denominator. The easiest way to do this, especially if the denominators are large, is usually to multiply the two denominators together and then reduce down if necessary.

Once you’ve found the lowest common denominator, then you have to multiply up the numerators to match.

Just as we reduced down the fractions in the previous section, now you have to multiply them up. As long as you always multiply or divide both top and bottom of a fraction by the same number, the **fraction remains the same**.

You therefore **multiply the numerator by whatever you had to multiply the denominator by to get to the LCD**.

Example 1

^{3}/_{5}+^{1}/_{6}

The smallest number that will divide by both denominators (5 and 6), is 30.

When you multiply 5 by 6, you also have to multiply 3 by 6 to get ^{18}/_{30}.

You had to multiply 6 by 5, so you now have to multiply 1 by 5, to get ^{5}/_{30}.

You now have a sum which looks like this:

^{18}/_{30} + ^{5}/_{30}

You can then add the two numerators together, 18 + 5 = 23.

The answer is therefore ^{23}/_{30}.

Example 2

^{3}/_{8}+^{1}/_{4}

Both 8 and 4 are factors of 8, so the LCD is 8.

You have not multiplied 8 by anything, so you do not need to change 3 either. You have multiplied 4 by 2, so you also need to multiply 1 by 2, to get 2.

Your sum now looks like this:

^{3}/_{8} + ^{2}/_{8}

The answer is therefore is ^{5}/_{8} .

Example 3

^{3}/_{4}-^{1}/_{2}

The LCD is 4, because 4 divides by 2.

The ^{1}/_{2} expressed as quarters is ^{2}/_{4}.

Your sum is therefore ^{3}/_{4} - ^{2}/_{4}

The answer is therefore is ^{1}/_{4} .

## Multiplying Fractions

See our page, Multiplication for more general help.

When multiplying fractions, you write the two fractions side by side.

Multiply the two numerators to get the answer numerator, and the two denominators to get the answer denominator.

Finally, reduce down the fraction to its smallest form.

Example 1

^{3}/_{5}×^{4}/_{7}

Multiply the numerators (top numbers) 3 × 4 = 12 and the denominators 5 × 7 = 35.

The answer is therefore ^{12}/_{35}

Example 2

^{2}/_{5}×^{5}/_{7}

Again, multiply the numerators 2 × 5 = 10 and the denominators 5 × 7 = 35.

This gives the answer ^{10}/_{35}

This time the fraction can be reduced as 10 and 35 are both divisible by 5.

The answer is therefore ^{2}/_{7}

## Dividing Fractions

See our page, Division for more general help.

To divide a fraction by another, turn the divisor fraction (the one that you are dividing by) upside down and then multiply (as above).

If this makes no sense, remember that multiplying by ^{1}/_{2} is the same as dividing by 2.

2 can be written as a fraction ^{2}/_{1}, so all you have done is turned the fraction upside down.

Example

^{3}/_{12}÷^{4}/_{7}

First turn the divisor fraction upside down and change the sum to a multiplication.

The sum is therefore ^{3}/_{12} × ^{7}/_{4}

Multiply the numerators 3 × 7 = 21 and the denominators 12 × 4 = 48.

This gives the answer ^{21}/_{48}

The fraction can be reduced as 21 and 48 are both divisible by 3.

The answer is therefore ^{7}/_{16}

### A Quick Note on Ratios

Ratios are another way to express fractions and decimals.

**A ratio of 1 in 5 is the same as a fraction of 1/5** or, expressed as a decimal, 0.2. All are ways of saying one part in five.

A ratio is generally written with a colon in the middle, so 1:5, 1:2 and so on.

Betting and Mathematics

The ‘odds’ for betting on racing, and indeed on anything else, are generally expressed as ratios. You will therefore see odds of 2-1, 11-7, and so on. In this case, the second number is what you stake, and the first is what you win.

For odds of 2-1, if you stake £1, you will win £2.

You may also see odds of 1-2 and evens. Evens means that the two numbers are the same. In betting terms, you will win what you staked.

Odds of 1-2 means that you stake £2 and win £1. Of course, you also get your stake back! Odds are sometimes taken as the bookmakers’ judgement of how likely that event is to occur. However, that’s not necessarily the case. Bookmakers, being businessmen and women, don’t want to lose money. Low odds usually mean that lots of people have placed a bet on that event, whether it’s a particular horse to win, or the sex of a royal baby.

The bookmakers don’t want to lose money, so they have reduced the possible payout. Sometimes, if too many people bet, the bookies will close the book altogether.

### To Conclude

At first glance, fractions may not look particularly useful.

However, when you think about dividing up a cake within a group, or even betting, you can see that fractions are vital to everyday life.

Learning how to manipulate fractions is a skill that will be useful in all kinds of circumstances.

Continue to:

Decimals

Percentages