Solving Multi-Step Ratio Problems Sequentially

Property In a multi step ratio problem, the solution to one proportion is used as a known value to set up and solve the next proportion: Step 1: Solve $$\frac{a}{b} = \frac{c}{x}$$ to find the intermediate value $x$. Step 2: Use $x$ as an input to solve the next proportion, such as $$\frac{x}{d} = \frac{y}{e}$$, to find the final unknown value $y$.

Examples Example 1: A car travels 60 miles on 2 gallons of gas. Gas costs 3 dollars per gallon. How much does it cost to travel 150 miles? Step 1 (Find gallons): $$\frac{60 \text{ miles}}{2 \text{ gallons}} = \frac{150 \text{ miles}}{x \text{ gallons}}$$, so $x = 5$ gallons. Step 2 (Find cost): $$\frac{3 \text{ dollars}}{1 \text{ gallon}} = \frac{y \text{ dollars}}{5 \text{ gallons}}$$, so $y = 15$ dollars. Example 2: Completing 4 game missions earns 20 points. It takes 50 points to earn 1 reward. How many missions must you complete to earn 3 rewards? Step 1 (Find points needed): $$\frac{50 \text{ points}}{1 \text{ reward}} = \frac{p \text{ points}}{3 \text{ rewards}}$$, so $p = 150$ points. Step 2 (Find missions): $$\frac{4 \text{ missions}}{20 \text{ points}} = \frac{m \text{ missions}}{150 \text{ points}}$$, so $m = 30$ missions.

Explanation Multi step ratio problems require you to break down a complex situation into a sequence of simpler proportional equations. First, identify the hidden intermediate value you need to bridge the gap between your starting information and your final goal. After solving the first proportion for this intermediate value, substitute it into a second proportion to find your final answer. Keeping careful track of your units at every step ensures that each proportion is set up correctly and your final result is reasonable.