Learn on PengiReveal Math, AcceleratedUnit 2: Proportional Relationships

Lesson 2-5: Use Proportional Reasoning to Solve Multi-Step Ratio Problems

In this Grade 7 lesson from Reveal Math, Accelerated (Unit 2: Proportional Relationships), students learn how to apply proportional reasoning to solve multi-step ratio problems by setting up and solving proportions across sequential steps. Using real-world contexts like Earth's rotation rate and video game scoring systems, students practice converting units, chaining proportions, and checking whether their solutions are reasonable.

Section 1

Solving Ratio Word Problems with Equivalent Ratios

Property

To solve a word problem using proportions, first identify the two quantities being compared and set up two equal ratios. Let a variable represent the unknown quantity. Ensure that the units are placed consistently in the numerators and denominators of the ratios.

unit A quantity 1unit B quantity 1=unit A quantity 2unit B quantity 2\frac{\text{unit A quantity 1}}{\text{unit B quantity 1}} = \frac{\text{unit A quantity 2}}{\text{unit B quantity 2}}

Examples

  • A doctor prescribes 5 ml5 \text{ ml} of medicine for every 25 lbs25 \text{ lbs} of weight. For a child weighing 80 lbs80 \text{ lbs}, we set up:
5 ml25 lbs=a80 lbs\frac{5 \text{ ml}}{25 \text{ lbs}} = \frac{a}{80 \text{ lbs}}

To solve, note that 8080 is 3.23.2 times 2525, so multiply 5 ml5 \text{ ml} by 3.23.2 to find a=16 mla = 16 \text{ ml}.

Section 2

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

ab=cx\frac{a}{b} = \frac{c}{x}
to find the intermediate value xx.
Step 2: Use xx as an input to solve the next proportion, such as
xd=ye\frac{x}{d} = \frac{y}{e}
, to find the final unknown value yy.

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):

60 miles2 gallons=150 milesx gallons\frac{60 \text{ miles}}{2 \text{ gallons}} = \frac{150 \text{ miles}}{x \text{ gallons}}
, so x=5x = 5 gallons.
Step 2 (Find cost):
3 dollars1 gallon=y dollars5 gallons\frac{3 \text{ dollars}}{1 \text{ gallon}} = \frac{y \text{ dollars}}{5 \text{ gallons}}
, so y=15y = 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):

50 points1 reward=p points3 rewards\frac{50 \text{ points}}{1 \text{ reward}} = \frac{p \text{ points}}{3 \text{ rewards}}
, so p=150p = 150 points.
Step 2 (Find missions):
4 missions20 points=m missions150 points\frac{4 \text{ missions}}{20 \text{ points}} = \frac{m \text{ missions}}{150 \text{ points}}
, so m=30m = 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.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Solving Ratio Word Problems with Equivalent Ratios

Property

To solve a word problem using proportions, first identify the two quantities being compared and set up two equal ratios. Let a variable represent the unknown quantity. Ensure that the units are placed consistently in the numerators and denominators of the ratios.

unit A quantity 1unit B quantity 1=unit A quantity 2unit B quantity 2\frac{\text{unit A quantity 1}}{\text{unit B quantity 1}} = \frac{\text{unit A quantity 2}}{\text{unit B quantity 2}}

Examples

  • A doctor prescribes 5 ml5 \text{ ml} of medicine for every 25 lbs25 \text{ lbs} of weight. For a child weighing 80 lbs80 \text{ lbs}, we set up:
5 ml25 lbs=a80 lbs\frac{5 \text{ ml}}{25 \text{ lbs}} = \frac{a}{80 \text{ lbs}}

To solve, note that 8080 is 3.23.2 times 2525, so multiply 5 ml5 \text{ ml} by 3.23.2 to find a=16 mla = 16 \text{ ml}.

Section 2

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

ab=cx\frac{a}{b} = \frac{c}{x}
to find the intermediate value xx.
Step 2: Use xx as an input to solve the next proportion, such as
xd=ye\frac{x}{d} = \frac{y}{e}
, to find the final unknown value yy.

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):

60 miles2 gallons=150 milesx gallons\frac{60 \text{ miles}}{2 \text{ gallons}} = \frac{150 \text{ miles}}{x \text{ gallons}}
, so x=5x = 5 gallons.
Step 2 (Find cost):
3 dollars1 gallon=y dollars5 gallons\frac{3 \text{ dollars}}{1 \text{ gallon}} = \frac{y \text{ dollars}}{5 \text{ gallons}}
, so y=15y = 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):

50 points1 reward=p points3 rewards\frac{50 \text{ points}}{1 \text{ reward}} = \frac{p \text{ points}}{3 \text{ rewards}}
, so p=150p = 150 points.
Step 2 (Find missions):
4 missions20 points=m missions150 points\frac{4 \text{ missions}}{20 \text{ points}} = \frac{m \text{ missions}}{150 \text{ points}}
, so m=30m = 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.