## Fractions

The top line of the fraction is called the numerator. The bottom line of the fraction is called the denominator.

### Types of Fractions

• A proper fraction is a fraction where the numerator is smaller than the denominator.
• e.g. 34 and 15 are proper fractions.
• An improper fraction is a fraction where the numerator is bigger than the denominator.
• e.g. 87 and 154 are improper fractions.
• A mixed number is an integer and a fraction written together.
• e.g. 312 and 114 are mixed numbers.
• Equivalent fractions represent the same number. They can be simplified or cancelled down to fractions with equal values.

e.g. { 12, 48, 918, 2754} are equivalent fractions because they can all be simplified to 12.

### Operations on Fractions

Always begin by changing any mixed numbers into improper fractions. If the calculation is very simple then the integers can be added together first, followed by the fractions.

e.g.  314 + 712 = 1034

• If the denominators are different, change them to equivalent fractions with the same denominator.
• Add or subtract the top lines only − never add or subtract the bottom lines!
• Simplify where possible.

 Examples Answers Calculate: (a) 3⁄5 + 4⁄5 3⁄5 + 4⁄5 = 7⁄5 = 12⁄5 (b) 8⁄9 − 5⁄6 8⁄9 − 5⁄6 = 16⁄18 − 15⁄18 = 1⁄18 (c) 13⁄4 + 22⁄5 13⁄4 + 22⁄5 = 65⁄20 + 88⁄20 = 153⁄20 = 713⁄20

Multiplication

• Step 1: Simplify by cancelling common factors between top and bottom lines.
• Step 2: Multiply the numerators together and then multiply the denominators together.

 Examples Answers Calculate: (a) 3⁄9 × 4⁄8 = 1⁄3 × 1⁄2 = 1⁄6 (by cancelling) (b) 31⁄2 × 13⁄7 = 31⁄2 × 13⁄7 = 7⁄2 × 10⁄7 = 1⁄1 × 5⁄1 = 5

Division

Turn the second fraction upside down (to make it into the reciprocal) and multiply.

 Examples Answers Calculate: (a)5 6 ÷ 5 12 5⁄6 ÷ 5⁄12 = 5⁄6 × 12⁄5 = 1⁄1 × 2⁄1 = 2 (b)51⁄4 ÷ 13⁄5 51⁄4 ÷ 13⁄5 = 21⁄4 × 5⁄8 = 105⁄32 = 39⁄32