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Math · 14 · Problem-Solving and Data Analysis

Ratio and Proportion

Notes

Key points

Ratio $a:b$ → amounts $ak$ and $bk$ ; total $(a+b)k$ .
Part-to-part vs part-to-whole: boys:girls 3:5 → boys are $\tfrac{3}{8}$ .
Proportion: same labels in the same positions, then cross-multiply.
Direct variation $y=kx$ : $\tfrac{y}{x}$ constant. Inverse: $xy$ constant.
Chain ratios via the shared term: $2:3$ and $4:5$ → $8:12:15$ .
Scale factor $k$ : lengths × $k$ , areas × $k^2$ , volumes × $k^3$ .

Formulas

Cross-multiplication

\dfrac{a}{b}=\dfrac{c}{d}\ \Rightarrow\ ad=bc

Share of a total

\text{part}=\dfrac{a}{a+b}\times\text{total}

Direct and inverse variation

y=kx\qquad y=\dfrac{k}{x}

Scaling

\text{area}\times k^2\qquad\text{volume}\times k^3

Common traps

Writing a 3:5 ratio as the fraction 3/5 of the whole.
Mixing units across a proportion (inches on one side, feet on the other).
Doubling the side and doubling the area — it quadruples.

SAT tip

Label each fraction in words (miles over hours = miles over hours) before cross-multiplying. Mismatched labels = wrong answer.

Test yourself

10 official SAT questions on Ratio and Proportion, easiest first.

Next lesson15 · Rates and Unit Conversion