Ratios, rates, and proportions

A ratio compares two quantities, and a proportion sets two ratios equal. Nearly every question here is solved by writing a proportion and cross-multiplying.

The one rule that prevents mistakes is consistency: whatever is on top in the first fraction must be on top in the second. Apples over price equals apples over price. Mixing the order is the commonest error and produces an answer that is offered.

A rate is a ratio with different units, such as kilometres per hour or dollars per kilogram. A unit rate has $1$ in the denominator, which makes comparisons easy — the better value is whichever has the lower cost per unit.

For a ratio shared between parts, such as splitting $\$60$ in the ratio $2:3$, add the parts to get $5$ shares, divide to find one share, then multiply.

Watch for questions giving a ratio and a difference rather than a total, since they need the same share method applied to the difference instead.

If $3$ apples cost $\$2$, what do $12$ apples cost?
Keep apples on top in both fractions

Split $\$60$ in the ratio $2:3$.
There are $2 + 3 = 5$ shares, so one share is $\$12$, giving $\$24$ and $\$36$.

Which is better value: $500$g for $\$4$ or $800$g for $\$6$?
Compare unit rates: $\frac{4}{500} = \$0.008$ per gram against $\frac{6}{800} = \$0.0075$ per gram. The $800$g pack is cheaper per gram.