GCSE Maths

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Fractions

What is a fraction?

The fraction "three over four" or "three fourths" or "three quarters" can be written as

	¾	3/4
The above fractions mean  3 ÷ 4

Definitions

In a fraction, the top number is called the numerator and the bottom number is called the denominator.

A proper fraction is a fraction where the top is less than the bottom, e.g. ⅞.

An improper fraction ('top-heavy' fraction) is a fraction where the top is larger than the bottom, e.g. ⁸/₇.

A mixed number has a whole number part followed by a proper fraction part, e.g. 4½ (which is the same as 4 + ½)

Example. Convert 4½ into an improper fraction.

4½ = (4 x 2) + 1 all divided by 2
     = ⁹/₂

Example. Express ⁷/₃ as a mixed number.

⁷/₃ = 2⅓ since 7 ÷ 3 = 2 remainder 1

Two fractions are said to be equivalent fractions if they have the same value, e.g . ⁶/₈ = ³/₄
If you multiply the top by a number and multiply the bottom by the same number, the fraction will remain the same i.e. equivalent.
Likewise if you divide the top by a number and divide the bottom by the same number, the fraction will remain the same i.e. equivalent.

Thus ⁶/₈ = ³/₄ because dividing both 6 and 8 in the first fraction by 2 gives you 3 over 4 (likewise multiplying both 3 and 4 in the second fraction by 2 gives you 6 over 8).

A fraction is said to be in its lowest terms when it is impossible to find a number that will divide exactly into both the top and the bottom. Thus ⁶/₈ is not in its lowest terms, whereas ³/₄ is.

The LCM (Lowest Common Multiple) of the denominators of several fractions is sometimes called the lowest common denominator.

Comparing fractions.

Express both fractions with the same bottom before comparing. In other words, the fractions should be 'same-denominator' fractions.

e.g. it is easier to compare ³/₈ and ²/₈ (same bottom) than it is to compare ³/₈ and ¹/₄ (different bottoms)

Adding and Subtracting

'Same-denominator' fractions can be added or subtracted by adding or subtracting their top numbers, and putting the result over the bottom number.

e.g. (a) ⅜ + ⅛ = ⁴/₈ = ½          (b) ⅜ - ⅛ = ²/₈ = ¼

If you are adding or subtracting fractions that do not have the same denominators, they should be converted to ones that do. (The resulting denominator should ideally be the lowest common denominator.)

e.g. ¹/₈ + ⁵/₆ = ³/₂₄ + ²⁰/₂₄ = ²³/₂₄

One way of adding or subtracting mixed fractions is to convert them to 'top-heavy' (i.e. improper) fractions first. Another way is to add/subtract the whole-number parts first and then do the same for the fractional parts.

e.g (a) 1⅜ + 3⅛ = (1+3) + (⅜+⅛) = 4 + ⁴/₈ = 4 + ½ = 4½

      (b) 1⅜ + 3⅞ = (1+3) + (⅜+⅞) = 4 + ¹⁰/₈ = 4 + ⁵/₄ = 4 + 1¼ = 5¼

Multiplying

First multiply the tops of the fractions and then multiply the bottoms. Then put one over the other.

e.g. ⅔ x ⅝ = 2x5 divided by 3x8 = ¹⁰/₂₄ = ⁵/₁₂ (reducing ¹⁰/₂₄ to its 'lowest terms'.)

Mixed fractions must be converted first to top-heavy (improper) fractions before multiplying.

Dividing

To divide one fraction by a second, turn the second upside-down and then multiply the two.
e.g. ²/₃ ÷ ¹/₄ = ²/₃ x ⁴/₁ = 2 x 4 divided by 3 x 1 = ⁸/₃

Powers

To raise a fraction to a power, raise both the top and the bottom to that power.

e.g. (⅔)³ = 2³ /3³ = 8/27

Square Roots

The square root of a fraction is equal to the square root of the top divided by the square root of the bottom.

Working out an expression involving fractions

• First work out whatever is inside a pair of brackets
• Multiply and/or divide
• Add and/or subtract