Factors and multiples

• Prime factorisation: write a number as a product of primes, e.g. $60 = 2^2 \times 3 \times 5$.
• $\text{GCD} \times \text{LCM} = \text{product of the two numbers}$ (for two numbers $a, b$: $\text{GCD}(a,b)\times\text{LCM}(a,b)=a\times b$).

Fractions

$$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} \qquad \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd} \qquad \frac{a}{b} \div \frac{c}{d} = \frac{a}{b}\times\frac{d}{c}$$

Percentages

$$\text{Percentage} = \frac{\text{part}}{\text{whole}} \times 100\% \qquad \text{Percentage change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100\%$$

• Increase by $r\%$: multiply by $\left(1 + \frac{r}{100}\right)$. Decrease by $r\%$: multiply by $\left(1 - \frac{r}{100}\right)$.
• Original amount $= \dfrac{\text{final amount}}{1 \pm \frac{r}{100}}$.

Squares, cubes and reciprocals

• $\sqrt{ab} = \sqrt{a}\sqrt{b}$, $\quad \sqrt{\dfrac{a}{b}} = \dfrac{\sqrt{a}}{\sqrt{b}}$
• Reciprocal of $x$ is $\dfrac{1}{x}$.