## What is a fraction?

So far we have dealt with whole numbers or integers, as found by counting.A fraction is one whole number divided by another.We write fractions as one number over another, with a horizontal line between them, like this:      3      –      4or in a line of text as one number with a slash or solidus then the other number, like this: 3/4.The fraction 3/4 or three quarters means 3 parts out of 4.The upper number, 3, is called the numerator and the lower number, 4, is the denominator.To calculate a fraction of something, multiply by the numerator and divide by the denominator.For example, 3/4 of 12 is 9:      3      – × 12 = (3 × 12) ÷ 4 = 36 ÷ 4 = 9       4

### Exercise: fractions

What is a half as a fraction?What is four fifths of 20?Check answer to fractions exercise.

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## Simplifying fractions

Sometime the numerator and denominator have a common factor. We can make the fraction simpler by dividing top and bottom by the factor.For example:      15     3 × 5         3      –– = –––––– = ––      20     4 × 5         4because the common factor of 15 and 20 is 5. We divide top and bottom by 5.

### Exercise: simplifying fractions

Simplify 27/30.Check answer to simplifying fractions exercise.

## Improper fractions

Fractions which are between 0 and 1 are called proper fractions.Improper fractions are those greater than 1.For example, 7/4 is an improper fraction.We can also express this as a whole number and a proper fraction: 1¾.

### Exercise: improper fractions

Express five and two thirds as an improper fraction.Express       17       ––         5as an integer and proper fraction.Check answer to improper fractions exercise.

## Reciprocals

The reciprocal of a number is one divided by the number.The reciprocal of 2 is ½.Conversely, the reciprocal of ½ is 2.To find the reciprocal of a fraction we turn it over, so that the numerator becomes the denominator and the denominator becomes the numerator.                                                 4           5 For example, the reciprocal of   ––   is   ––                                                  5           4 We can convert this to an integer and a proper fraction as 1¼.

### Exercise: reciprocals

What are the the reciprocals of 4 and of 5/6?Check answer to reciprocals exercise.

## Multiplying fractions

To multiply two fractions we simply multiply their numerators,multiply their denominators, and simplify if necessary:      2       1          2 × 1           2           1       –– × ––   =   –––––   =   ––   =   ––       3       2          3 × 2          6          3

### Exercise: multiplying fractions

Multiply four fifths by three quarters.Check answer to multiplying fractions exercise.

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