Fractions

Spec reference: Number - Fractions, decimals and percentages
Key idea: Simplify, compare and calculate with fractions including mixed numbers.

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Key vocabulary

• Numerator - the top number
• Denominator - the bottom number
• Proper fraction - numerator < denominator, e.g. ${\displaystyle \frac{3}{5}}$
• Improper fraction - numerator > denominator, e.g. ${\displaystyle \frac{7}{3}}$
• Mixed number - whole number + fraction, e.g. $2{\displaystyle \frac{1}{3}}$

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Simplifying fractions

Divide numerator and denominator by their HCF.

Example

Simplify ${\displaystyle \frac{18}{24}}$

HCF of 18 and 24 is 6:

$$\frac{18}{24}=\frac{18÷6}{24÷6}=\boxed{{\displaystyle \frac{3}{4}}}$$

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Converting between improper fractions and mixed numbers

Improper to mixed number: divide numerator by denominator.

Example

Convert ${\displaystyle \frac{17}{5}}$ to a mixed number.

$17÷5=3$ remainder $2$

$$\frac{17}{5}=\boxed{{\displaystyle 3\frac{2}{5}}}$$

Mixed number to improper fraction: multiply whole number by denominator, add numerator.

Example

Convert $4{\displaystyle \frac{2}{3}}$ to an improper fraction.

$$4\frac{2}{3}=\frac{(4\times 3)+2}{3}=\boxed{{\displaystyle \frac{14}{3}}}$$

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Adding and subtracting fractions

Fractions must have the same denominator before adding or subtracting.

Example

Calculate ${\displaystyle \frac{2}{3}}+{\displaystyle \frac{3}{4}}$

LCM of 3 and 4 is 12:

$$\frac{2}{3}+\frac{3}{4}=\frac{8}{12}+\frac{9}{12}=\frac{17}{12}=\boxed{{\displaystyle 1\frac{5}{12}}}$$

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Multiplying fractions

Multiply numerators together and denominators together.

$$\frac{a}{b}\times \frac{c}{d}=\frac{a\times c}{b\times d}$$

Example

${\displaystyle \frac{2}{3}}\times {\displaystyle \frac{4}{5}}={\displaystyle \frac{2\times 4}{3\times 5}}={\displaystyle \frac{8}{15}}$

Cancel first

Look for common factors between any numerator and any denominator before multiplying - it keeps numbers smaller.

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Dividing fractions

Keep, Change, Flip (KCF): keep the first fraction, change $÷$ to $\times$, flip the second fraction.

$$\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}\times \frac{d}{c}$$

Example

${\displaystyle \frac{3}{4}}÷{\displaystyle \frac{2}{5}}={\displaystyle \frac{3}{4}}\times {\displaystyle \frac{5}{2}}={\displaystyle \frac{15}{8}}=\boxed{{\displaystyle 1{\displaystyle \frac{7}{8}}}}$

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Fraction of an amount

Multiply the amount by the fraction.

Example

Find ${\displaystyle \frac{3}{8}}$ of 56.

$$\frac{3}{8}\times 56=\frac{3\times 56}{8}=\frac{168}{8}=\boxed{{\displaystyle 21}}$$

Or: $56÷8=7$, then $7\times 3=21$

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Exam tips

Watch out for

• Never add the denominators together: ${\displaystyle \frac{1}{3}}+{\displaystyle \frac{1}{4}}\ne {\displaystyle \frac{2}{7}}$
• Always simplify your final answer if possible
• With mixed numbers in multiplication/division, convert to improper fractions first

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Test Yourself

Question 1 of 5

Simplify $\frac{24}{36}$