Lesson 54: Rate Word Problems

New Concept

This course teaches you to translate real-world situations into mathematical language, building a powerful toolkit for solving any quantitative problem you encounter.

What’s next

This is the big picture for the entire course. To begin, we will apply these skills to solve rate word problems using ratio boxes.

Property

To find the distance traveled, set up a proportion comparing the rate of speed to the actual measures of distance and time.

Examples

• A scooter travels at 25 miles per hour. How far will it go in 3 hours?
  $$\frac{25}{1} = \frac{d}{3} \rightarrow 1 \times d = 25 \times 3 \rightarrow d = 75 \text{ miles}$$
• A train moves at 80 miles per hour. How far does it travel in 1.5 hours?
  $$\frac{80}{1} = \frac{d}{1.5} \rightarrow 1 \times d = 80 \times 1.5 \rightarrow d = 120 \text{ miles}$$

Explanation

Think of your car's speed as a fixed rate. A ratio box helps you organize this information to create a simple proportion. This lets you scale up from the one-hour rate to find the total distance for any amount of time. It's a journey-planning superpower!

Property

To find how much fuel is needed for a trip, use a proportion that relates fuel efficiency (miles per gallon) to the total distance and total fuel:

Examples

• A car gets 30 miles per gallon. How many gallons are needed for a 360-mile trip?
  $$\frac{30}{1} = \frac{360}{g} \rightarrow 30g = 360 \rightarrow g = 12 \text{ gallons}$$
• If a truck averages 20 miles per gallon, how much fuel is used for 500 miles?
  $$\frac{20}{1} = \frac{500}{g} \rightarrow 20g = 500 \rightarrow g = 25 \text{ gallons}$$

Explanation

Your car's miles per gallon (MPG) is a rate! A ratio box lets you set up a simple proportion to figure out exactly how much fuel you need for a road trip. This way, you can plan your stops and budget for gas without any guesswork.

Property

Find an hourly rate of pay by setting up a proportion that compares the amount earned to the time worked:

Examples

• If you earn 80 dollars for 10 hours of work, what is your hourly rate?
  $$\frac{p}{1} = \frac{80}{10} \rightarrow 10p = 80 \rightarrow p = 8 \text{ dollars per hour}$$
• Using a rate of 15 dollars per hour, how much would you earn in 40 hours?
  $$\frac{15}{1} = \frac{T}{40} \rightarrow 1 \times T = 15 \times 40 \rightarrow T = 600 \text{ dollars}$$

Explanation

Wondering what you earn each hour? A ratio box makes it a piece of cake. Just compare your total pay to the hours you worked to find the rate. Once you have that magic number, you can easily calculate your earnings for any amount of work time!