Solving Proportions

Solving Proportions

A proportion is an equation stating that two ratios are equal, such as \frac{a}{b} = \frac{c}{d}. Proportions let us solve for an unknown quantity whenever we know that two ratios must match.

Cross-Multiplication

The fastest tool for solving a proportion is cross-multiplication: if \frac{a}{b} = \frac{c}{d}, then a \cdot d = b \cdot c. This works because multiplying both sides of the equation by bd clears both denominators at once. For example, to solve \frac{x}{6} = \frac{10}{15}, cross-multiply to get 15x = 60, so x = 4.

Scaling

Sometimes it's faster to notice a direct scale factor between the two ratios instead of cross-multiplying. In \frac{4}{9} = \frac{x}{45}, notice that 45 = 9 \times 5, so the scale factor is 5; applying it to the numerator gives x = 4 \times 5 = 20. Watching for a clean scale factor like this is often much faster than cross-multiplying on a timed contest.

Similar Figures

Two figures are similar when they have the same shape but possibly different sizes — all corresponding angles are equal, and all corresponding side lengths are in the same ratio. If two triangles are similar with a scale factor of \frac{3}{2} between corresponding sides, then every pair of corresponding sides shares that same ratio. This turns geometry problems about similar figures into proportion problems: set corresponding sides equal as a ratio, then cross-multiply or scale to find an unknown length.

Setting Up Proportions Correctly

The most common proportion mistake is mismatching what goes in the numerator and denominator. Always keep the same type of quantity in the same position on both sides. If a map shows 2 inches representing 50 miles, and you want to know how many miles 7 inches represents, set it up as \frac{\text{inches}}{\text{miles}} on both sides: \frac{2}{50} = \frac{7}{m}, giving 2m = 350, so m = 175 miles. Flipping the setup (miles over inches on one side, inches over miles on the other) is a common source of errors — always double check the units line up.

Example

Two similar rectangles have a length ratio of 5:8. If the smaller rectangle has width 15, and the rectangles' widths are also in the same ratio, find the width of the larger rectangle.

Step 1: Set up the proportion, keeping "smaller" and "larger" consistent: \frac{5}{8} = \frac{15}{w}

Step 2: Cross-multiply: 5w = 8 \times 15 = 120.

Step 3: Solve: w = \frac{120}{5} = 24.

The larger rectangle's width is 24.

Key Takeaways

• A proportion sets two ratios equal; cross-multiplication (a/b = c/d \Rightarrow ad = bc) is the standard solving tool.
• Watch for an obvious scale factor between the two ratios — it's often faster than cross-multiplying.
• Similar figures have proportional corresponding sides — set up a proportion between matching sides to find unknown lengths.
• Always keep the same quantity type in the same position (numerator or denominator) on both sides of a proportion.