A ratio expresses how two or more quantities compare with each other. Ratios are used in many contexts: mixing, scaling, map reading, and probability.
Key skills:
- Simplify a ratio by dividing all parts by the HCF
- Convert between ratios and fractions
- Divide an amount in a given ratio
- Use ratio to find an unknown when one part is given
Example — find an unknown part: Two numbers are in the ratio $3:5$. The smaller number is 12. Find the larger. $$\frac{12}{3} = 4 \quad \text{(one share)} \implies \text{larger} = 5 \times 4 = 20$$
Example — ratio to fraction: In a class, boys to girls = $2:3$. $$\text{Fraction that are boys} = \frac{2}{5}$$
Ratios must compare quantities in the same units. Convert first if needed (e.g. 30 minutes : 2 hours = 30 : 120 = 1 : 4).
Common error: treating a ratio $a:b$ as $\frac{a}{b}$ rather than $\frac{a}{a+b}$ when finding the fraction of a total.