Proportion
A proportion is a statement that two ratios are equal, a/b = c/d, and the workhorse move is cross-multiplication: a·d = b·c. That turns a proportion into a linear equation you can solve for any missing term — the basis of scaling recipes, reading maps, and comparing similar figures. In a proportion a : b = c : d, the outer terms a and d are the extremes and the inner terms b and c are the means, and the product of the means equals the product of the extremes. The mean proportional (or geometric mean) between a and b is √(ab), which the GRE sometimes tucks into a Quantitative Comparison. When you set up a proportion from a word problem, keep the units aligned across the equals sign — miles over hours must equal miles over hours — because a flipped ratio is the most common way to get a proportion wrong.
✅ Solved examples
✏️ Practice — try these, take hints as needed
📝 Topic test — 8 questions
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Formula Reference Sheet
Ratio basics
| Ratio as a fraction | a : b = a/b (same information) |
|---|---|
| Scaling a ratio | a : b = ka : kb for any k ≠ 0 |
| Dividing a total in a : b | shares = [a/(a+b)] × T and [b/(a+b)] × T |
| Combining a : b and b : c | make the shared term equal, then read a : b : c |
Proportion & variation
| Proportion (cross-multiply) | a/b = c/d ⇔ a·d = b·c |
|---|---|
| Direct variation | y = kx ⇒ y₁/x₁ = y₂/x₂ (ratio constant) |
| Inverse variation | y = k/x ⇒ x₁y₁ = x₂y₂ (product constant) |
| Mean proportional of a and b | √(ab) |