Arithmetic/Fractions

Fractions
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: $$ \frac{3}{5} $$

or 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
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
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


 * $$ \frac{4}{6} = \frac{4/2}{6/2} = \frac{2}{3}$$.

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
Sometimes you may meet 'mixed fractions' like $$1\frac{3}{5}$$. This really means $$1+\frac{3}{5}$$, or in words one and three-fifths. Mixed fractions are hard to work with... how would you go about working out the following?


 * $$1\frac{3}{5} \times 2\frac{2}{7}$$

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

$$1\frac{3}{5}=1+\frac{3}{5}=\frac{5}{5}+\frac{3}{5}=\frac{8}{5}\,\,\,\,$$

and

$$2\frac{2}{7}=2+\frac{2}{7}=\frac{14}{7}+\frac{2}{7}=\frac{16}{7}$$

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

$$1\frac{3}{5} \times 2\frac{2}{7}=\frac{8}{5} \times \frac{16}{7}=\frac{128}{35}=\frac{105}{35}+\frac{23}{35}=3\frac{23}{35}$$

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
 {Which of the following is the same as $$3\frac{3}{5}$$?} -5/18 -9/5 -18/3 +18/5 -11/5