Add / subtract $\dfrac{a}{b}\pm\dfrac{c}{d}=\dfrac{ad\pm bc}{bd}$make the denominators the same first, then $\pm$ the numerators
Multiply $\dfrac{a}{b}\times\dfrac{c}{d}=\dfrac{ac}{bd}$multiply across the tops and across the bottoms (cancel first if you can)
Divide $\dfrac{a}{b}\div\dfrac{c}{d}=\dfrac{a}{b}\times\dfrac{d}{c}$"keep, flip, multiply": use the reciprocal of the second fraction
Fraction of an amount $\dfrac{a}{b}\text{ of }N=(N\div b)\times a$divide by the bottom, multiply by the top
Subtract · $\dfrac{3}{4}-\dfrac{2}{3}$
common denominator $12$: $\dfrac{9}{12}-\dfrac{8}{12}$
$=\dfrac{1}{12}$
Divide · $\dfrac{2}{3}\div\dfrac{4}{5}$
keep, flip, multiply: $\dfrac{2}{3}\times\dfrac{5}{4}$
$=\dfrac{10}{12}=\dfrac{5}{6}$
Fraction of an amount · $\dfrac{3}{5}$ of $£20$
$20\div 5=4$
$4\times 3=£12$
Numerator: the top number – how many parts you have.
Denominator: the bottom number – how many equal parts the whole is split into.
Improper fraction: numerator $\ge$ denominator, so it is $1$ or more, e.g. $\tfrac{7}{4}$ ("top-heavy").
Mixed number: a whole number and a proper fraction together, e.g. $1\tfrac{3}{4}$.
Reciprocal: a fraction turned upside down; $\tfrac{a}{b}\to\tfrac{b}{a}$. Dividing uses it.
✗ Adding the denominators: $\tfrac{1}{2}+\tfrac{1}{3}=\tfrac{2}{5}$
✓ find a common denominator first: $\tfrac{3}{6}+\tfrac{2}{6}=\tfrac{5}{6}$.
✗ Flipping when multiplying (or flipping the first fraction)
✓ only flip when dividing, and only the second fraction.
✗ Leaving the answer un-simplified, e.g. $\tfrac{6}{8}$
✓ always cancel to lowest terms: $\tfrac{6}{8}=\tfrac{3}{4}$.
✗ Multiplying mixed numbers as wholes then fractions
✓ change mixed numbers to improper fractions before $\times$ or $\div$.
• To add or subtract, the denominators must be the same; to multiply or divide, they can be different.
• Multiplying by a fraction less than $1$ makes a number smaller; dividing by one less than $1$ makes it bigger.
• Any whole number is a fraction over $1$, e.g. $5=\tfrac{5}{1}$ – useful when dividing.
• For $\times$ and $\div$, turn any mixed number into an improper fraction first.
• Divide $=$ "keep, flip, multiply" (keep the first, flip the second, multiply).
• Cancel before multiplying to keep numbers small, and always give the answer in its lowest terms.
mathedup.co.uk · sheet Q65G