A **fraction** divides an item into smaller, equal, parts. Divide a circle into three equal parts: all three slices together will make a whole circle, but the circle itself has been divided by 3.

As a fraction, this is written as `frac(1)(3)`, which is another way of writing 1 ÷ 3. The top number is the **numerator**, and the bottom number is the **denominator**.

Two slices of the circle will be 2 x `frac(1)(3)`, and is written as `frac(2)(3)` (two lots of `frac(1)(3)`).

If each original slice is further divided in two, there will be six slices: each slice will be `frac(1)(6)` of the circle.

Two of the divided slices will be the same size as one of the original slices: `frac(2)(6)` = `frac(1)(3)`. This is an **equivalent fraction**: two fractions are equivalent if they have the same *value*, even if they are composed of different numbers.

Equivalent fractions can be obtained by multiplying or dividing the numerator and denominator by the same number:

`frac(1^(times2))(4_(times2)) = frac(2^(times2))(8_(times2)) = frac(4)(16)`

Dividing numerators and denominators by the same number will make them smaller. This is called **simplifying**. When the fraction cannot be simplified any more, it is in its **simplest form**.

`frac(6^(÷3))(12_(÷3)) = frac(2^(÷2))(4_(÷2)) = frac(1)(2)`

A **common fraction**, sometimes also called a **vulgar fraction**, is a fraction where both the numerator and the denominator are integers.

Put `frac(9)(27)` into its simplest form.

Divide both the numerator and denominator by 9.

`frac(9)(27) = frac(9^(÷9))(27_(÷9)) = frac(1)(3)`

It is also possible to divide top and bottom by 3, then to divide by 3 again, to get the same answer.

Answer: `frac(1)(3)`

Which of the following fractions are equivalent to `frac(3)(8)`?

`frac(1)(4)` , `frac(6)(16)` , `frac(8)(3)` , `frac(12)(32)` , `frac(3)(4)`

`frac(6)(16)` is `frac(3^(xx2))(8_(xx2))` and `frac(12)(32) = frac(3^(xx4))(8_(xx4))`

Answer: `frac(6)(16)` and `frac(12)(32)`

See also Terminating Decimals and Fractions

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