# Arithmetic/Fractions

## Fractions[edit | edit source]

A **fraction** consists of one quantity divided by another quantity. The fraction "three divided by five" or "three over five" or "three fifths" can be written as:

**3/5**

In this section, we will use the notation 3/5. Note that some fractions have special names.

- 1/2 - Instead of calling this a “twoth”, or a ”second”, it's called a
*half*. - 3/4 - As well as three
*fourths*, it can also be called three*quarters*.

### Numerator and Denominator[edit | edit source]

The first quantity, the number on top of the fraction, is called the **numerator**. It tells you the number, or how many of something you've got. The other number, on the bottom, is called the **denominator**. The denominator tells you the *denomination* of the fraction, which is really just a fancy way of telling you the *type* of the fraction. In exactly the same way, we know a £5 note and a £10 pound note are different because they are different types.

So, look at a number like 3/5. The numerator is 3. So whatever we've got, we've got three of them. The denominator is 5, so we've got fifths, whatever they are. Put the two together, we've got *three* of those things called *fifths*. We've got three fifths, we've got 3/5. *Same value, different appearance*.

Several rules for calculation with fractions are useful:

## Changing the *type* of a fraction[edit | edit source]

If both the numerator and the denominator of a fraction are multiplied or divided by the same number, then the fraction does not change its value.

This means we can make some fractions simpler, by making the numbers involved smaller. When the numbers are as small as possible, the fraction is said to be expressed in its lowest terms, or reduced. In a reduced fraction, the numerator and denominator are not divisible do not share any common divisors greater than one. Let's look at an example:
4/6 = 2/3.

Remember what that ' = ' sign means? It means that 4/6 and 2/3, although they look different, have the same value. *Same value, different appearance*.

This is because

- .

Try it out yourself with a calculator or with long division. From here, there is nothing we can divide both numbers by, so the fraction is reduced, which is our goal. To find out what to divide them by, you need to find a *common divisor*, which is a number they are both divisible by.

## Mixed fractions[edit | edit source]

Sometimes you may meet 'mixed fractions' like . This really means , or in words *one and three-fifths*. Mixed fractions are hard to work with... how would you go about working out the following?

The answer is to convert the mixed fraction into a standard fraction of the form numerator/denominator. For example,

and

Now, it's simple to multiply the fractions together:

Keep in mind that with mixed fractions, the whole number represents the *quotient* whereas the fraction represents the *remainder*. If you were to divide 25 by 5, you would be left with only 5 (and no remainder). However, if you were to divide 27 by 5, you would be left with 5 2/5.

## Exercises[edit | edit source]