Proportion Word Problems

A proportion is an equation stating that two ratios (or fractions) are equal. In the real world, proportions help us scale things up or down, such as adjusting a recipe, reading a map, or calculating distances.

Setting Up a Proportion

The most important rule for solving proportion word problems is to keep your units consistent.

If you put the unit "centimeters" on top and "kilometers" on the bottom for your first ratio, you must do exactly the same for your second ratio:

$\frac{\text{centimeters}}{\text{kilometers}} = \frac{\text{centimeters}}{\text{kilometers}}$

Solving by Cross-Multiplying

Once your proportion is set up, you can solve for the missing value using cross-multiplication. If you have a proportion in the form:

$\frac{a}{b} = \frac{c}{d}$

You can multiply diagonally to get a simple equation:

$a \times d = b \times c$

Example 1: Map Distances

Problem: On a map, $2\text{ cm}$ represents $50\text{ km}$. How far in real life is a distance of $5\text{ cm}$ on the map?

Step 1: Set up the proportion. Match the units on both sides (cm over km): $\frac{2}{50} = \frac{5}{x}$

Step 2: Cross-multiply. Multiply the diagonals: $2 \times x = 50 \times 5$ $2x = 250$

Step 3: Solve for $x$. Divide both sides by 2: $x = 125$

Answer: The actual distance is $125\text{ km}$.

Example 2: Recipe Adjustments

Problem: A recipe for $4$ people needs $3$ cups of flour. How much flour is needed to make the recipe for $10$ people?

Step 1: Set up the proportion. Match the units (people over cups): $\frac{4}{3} = \frac{10}{x}$

Step 2: Cross-multiply. $4 \times x = 3 \times 10$ $4x = 30$

Step 3: Solve for $x$. Divide both sides by 4: $x = \frac{30}{4} = 7.5$

Answer: You will need $7.5$ cups of flour.