Solving Ratio and Rate Problems

Lesson Objectives

• Apply ratio reasoning to solve real-world problems
• Use tape diagrams to represent and solve ratio problems
• Understand and calculate rates and unit rates
• Apply rates and unit rates to solve real-world problems
• Introduce slope as a rate in mathematical contexts

Common Core Standards

6.RP.A.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship.

6.RP.A.3b Solve unit rate problems including those involving unit pricing and constant speed.

Prerequisite Skills

• Understanding of ratios and equivalent ratios
• Basic problem-solving skills
• Division skills

Key Vocabulary

• Tape diagram
• Proportion
• Scale factor
• Rate
• Unit rate
• Rate of change

Warm-up Activity (5 minutes)

Present two problems on the board:

1. It cost $35 to fill a car's 10-gallon tank. What is the cost of gas per gallon?
2. A cyclist travels 24 miles in 2 hours. What is the cyclist's speed in miles per hour?

Ask students to discuss in pairs how they might approach solving these problems.

Teach (25 minutes)

Definitions

Use this slide show to introduce key definitions:

https://www.media4math.com/library/slideshow/definitions-ratios-and-rates

Instruction

Tape Diagrams. Introduce tape diagrams as another tool for solving ratio problems.

Demonstrate how to use tape diagrams for more complex rate problems. Use the scenario shown below, or use the following slide show, which represents the situation:

https://www.media4math.com/library/slideshow/using-tape-diagrams

• Example 1: A recipe calls for 2 cups of flour and 3 eggs to make 12 muffins. How much flour and how many eggs are needed to make 36 muffins?
  • Draw a tape diagram that shows the ratio of flours to eggs
  • This ratio results in 12 muffins.
  • To increase to 36 muffins means to scale the recipe by a factor of 3.
  • Calculate: 2 cups flour × 3 = 6 cups flour, and 3 eggs × 3 = 9 eggs
• Example 2: For every 3 muffins baked, a bakery makes 4 cookies. If 48 muffins are baked, how many cookies are also baked?
  • Draw a tape diagram that shows the ratio of muffins to cookies.
  • Write 48 to the right of the muffins part of the tape diagram.
  • Find the scaling factor to go from 3 to 48.
  • Use this scaling factor to find the corresponding number of cookies.

Rates. Use this slide show to introduce ratios, rates, and unit rates:

https://www.media4math.com/library/slideshow/introduction-rates

Next, use this video to show examples of calculating rates:

https://www.media4math.com/library/1796/asset-preview

Focus on an example using hourly wage:

• Explain how to calculate hourly wage as a rate (earnings per hour) using an example: If someone earns \$96 for working 8 hours, their hourly wage is \$96 ÷ 8 = \$12 per hour.

Show how hourly wage can be used to calculate wages for any amount of hours:

• Let w = hourly wage and h = number of hours worked
• Total earnings = w × h
• For example, if w = \$12/hour and h = 20 hours, total earnings = \$12 × 20 = \$240

Rate of change. Explain rates and unit rates in various contexts:

• Speed (miles per hour)
• Pricing (cost per item or group of items)
• Productivity (tasks per hour)
• Hourly wage (dollars per hour)
• Slope (rise over run)

Show how to calculate and apply rates and unit rates using real-world examples:

• If a car travels 240 miles in 4 hours, what is its speed in miles per hour?
• If an employee earns $420 for a 35-hour work week, what is their hourly wage?
• Introduce slope as a rate: If a line rises 3 units for every 4 units it runs horizontally, what is its slope?

Connect rate and slope of a linear function. Here is an example:

A taxi charges \$0.20 per mile with an initial charge of $5. Develop a linear function. Graph it and use it to find the cost of a 25-mile fare.

y = 0.2x + 5

Here is a Desmos activity that you can use to explore different fares:

https://www.desmos.com/calculator/6dfq3qc22y

Reinforce that the slope is the rate per mile.

Review (10 minutes)

Divide the class into small groups. Provide each group with a set of ratio, rate, and unit rate problems to solve using tape diagrams and rate calculations. Include problems involving hourly wages and a simple slope problem. Encourage students to explain their reasoning to each other.

Example problems:

1. A recipe calls for 2 cups of flour for every 3 cups of sugar. If you use 8 cups of flour, how much sugar do you need?
2. If an employee earns $585 for a 45-hour work week, what is their hourly wage?
3. A line rises 6 units as it runs 2 units to the right. What is its slope?

Circulate among the groups, offering guidance and clarification as needed.

Assess (5 minutes)

Administer a 10-question quiz to assess understanding. Students should complete this individually.

Quiz

1. If the ratio of dogs to cats at a pet store is 5:3, and there are 18 cats, how many dogs are there?
   
    
2. A car travels 180 miles in 3 hours. What is its speed in miles per hour?
   
    
3. If 4 notebooks cost $10, what is the price of one notebook?
   
    
4. In a bag of marbles, the ratio of red to blue marbles is 2:7. If there are 63 blue marbles, how many red marbles are there?
   
    
5. A printer can print 30 pages in 5 minutes. How many pages can it print in 1 minute?
   
    
6. If 15 oranges weigh 5 pounds, what is the weight of one orange in ounces? (1 pound = 16 ounces)
   
    
7. A recipe uses 3 eggs for every 2 cups of flour. If you want to use 10 eggs, how many cups of flour do you need?
   
    
8. If 8 gallons of gas cost $24, what is the cost of 12 gallons?
   
    
9. An employee earns $384 for a 32-hour work week. What is their hourly wage?
   
    
10. A line rises 9 units as it runs 3 units to the right. What is its slope?
    
     

Answers

1. 30 dogs
2. 60 miles per hour
3. $2.50
4. 18 red marbles
5. 6 pages per minute
6. 5 1/3 ounces
7. 6 2/3 cups of flour
8. $36
9. $12 per hour
10. 3 (slope = rise/run = 9/3 = 3)

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