Basic Fractions

A fraction is part of a whole
A fraction is written as $\frac{a}{b}$ where a and b are whole numbers (integers)

“Splitting something into 2 parts and taking 1 of these parts” gives the same overall amount as “splitting something into 4 parts and taking 2 of these parts”
- $\frac{1}{2}$ is the same as $\frac{2}{4}$
- Using the same idea, $\frac{2}{4}$ is the same as $\frac{4}{8}$ , or $\frac{8}{16}$ , or $\frac{3}{6}$ etc
Equivalent fractions are two fractions that represent the same amount (they are different ways of writing the same thing)
To make equivalent fractions, multiply the top and bottom of a fraction by the same amount
- $\frac{5}{6}$ is equivalent to $\frac{5 \times 2}{6 \times 2} = \frac{10}{12}$ and $\frac{5 \times 3}{6 \times 3} = \frac{15}{18}$ and $\frac{5 \times 4}{6 \times 4} = \frac{20}{24}$ …etc
- For each fraction, there are an infinite number of equivalent fractions
To cancel a fraction down to its “simplest” equivalent form, divide the top and bottom of a fraction by the biggest whole number that goes into both (“common factor”)
- $\frac{12}{18} = \frac{12 \div 6}{18 \div 6} = \frac{2}{3}$
- $\frac{25}{45} = \frac{25 \div 5}{45 \div 5} = \frac{5}{9}$

Method 1: divide by the denominator and multiply by the numerator
- To find $\frac{1}{2}$ of an amount, divide it by 2
- To find $\frac{1}{3}$ of an amount, divide it by 3
- To find $\frac{2}{5}$ of an amount, divide it by 5 then multiply it by 2
  - $\frac{2}{5}$ of 60: do 60 ÷ 5 = 12, then 12 × 2 = 24
Method 2 (if you know how to change fractions into decimals): change the fraction into a decimal, then multiply
- To find $\frac{1}{4}$ of an amount, multiply the amount by 0.25
- To find $\frac{9}{10}$ of an amount, multiply the amount by 0.9
Method 3 (if you know how to multiply fractions): write both numbers as fractions and multiply two fractions together
- To find $\frac{2}{3}$ of 60, work out $\frac{2}{3} \times \frac{60}{1}$
- To find $\frac{5}{6}$ of $\frac{8}{3}$ , work out $\frac{5}{6} \times \frac{8}{3}$

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Mixed Numbers & Top Heavy Fractions

A mixed number has a whole number (integer) part and a fraction part
eg. $3 \frac{3}{4}$ has the whole number 3 and the fraction $\frac{3}{4}$ , meaning “three and three quarters”
A top heavy fraction – also called an improper fraction – is one with the top (numerator) bigger than the bottom (denominator)
eg. $\frac{15}{4}$ means “fifteen quarters”

(a) Write $5 \frac{3}{4}$ as an improper fraction

Multiply the whole number by the denominator, and add to the numerator. Keep the denominator the same.

$\frac{(5 \times 4) + 3}{4}$

Simplify

$\frac{23}{4}$

(b) Write $\frac{17}{5}$ as a mixed number

Divide the top by the bottom

17 ÷ 5 = 3 remainder 2

The final answer is 3, with 2 parts still left over to be divided by 5, which can be written as a mixed number

$3 \frac{2}{5}$