Solving Ratio and Rate Problems

Lesson Summary

In this lesson, students will delve into the concepts of ratios and rates, understanding their definitions, representations, and applications in real-world scenarios. The lesson emphasizes the distinction between ratios and rates, introduces unit rates, and demonstrates how to solve problems involving these concepts using various strategies, including tape diagrams and proportional reasoning.

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

• Ratio: A comparison of two quantities that indicates their relative sizes. Ratios can be expressed in various forms, such as "a to b," "a:b," or as a fraction a/b. For example, the ratio of 2 apples to 3 oranges can be written as 2:3 or 2/3.
  • Multimedia Resource: https://www.media4math.com/library/22157/asset-preview
• Rate: A specific type of ratio that compares two quantities with different units. For instance, if a car travels 150 miles in 3 hours, the rate is 150 miles per 3 hours, or 50 miles per hour.
  • Multimedia Resource: https://www.media4math.com/library/22156/asset-preview
• Unit Rate: A rate in which the second quantity is one unit. It describes how many units of the first quantity correspond to one unit of the second quantity. Using the previous example, the unit rate is 50 miles per hour, meaning the car travels 50 miles in one hour.
  • Multimedia Resource: https://www.media4math.com/library/43395/asset-preview
• Tape Diagram: A visual tool that uses rectangles to represent the parts of a ratio, helping to illustrate the relationship between quantities and solve ratio-related problems.
• Proportion: An equation that states that two ratios are equivalent. For example, if 2/3 = 4/6, then 2:3 is proportional to 4:6.
  • Multimedia Resource: https://www.media4math.com/library/43387/asset-preview
• Scale Factor: A number used to multiply or divide quantities in a ratio to produce an equivalent ratio. For instance, multiplying both terms of the ratio 2:3 by 2 yields the equivalent ratio 4:6.
  • Multimedia Resource: https://www.media4math.com/library/43390/asset-preview
• Rate of Change: A measure that describes how one quantity changes in relation to another. In the context of a graph, it is often referred to as the slope, representing the change in the y-value per unit change in the x-value.

Multimedia Resources

• A collection of definitions on the topic of ratios, proportions, and percents: https://www.media4math.com/Definitions--RatiosProportionsPercents
• A student tutorial slide show on definitions on the topic of ratios, proportions, and percents: https://www.media4math.com/library/slideshow/student-tutorial-ratios-proportions-and-percents-definitions

Warm Up Activities

Choose from one ore more of these activities.

Activity 1: Review of Rates

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.

Activity 2: Brief Review of Slope

Before introducing rates and unit rates, students will review the concept of slope as a type of ratio to describe how one quantity changes in relation to another.

Instructions:

1. Write the following points on the board and ask students to determine the slope between each pair:
   • $(1,2)$ and $(3,6)$
   • $(2,4)$ and $(5,10)$
   • $(0,0)$ and $(4,8)$
2. Guide students to recall the slope formula: $$m = \frac{\text{change in } y}{\text{change in } x} = \frac{y_2 - y_1}{x_2 - x_1}$$
3. Have students calculate the slope for each pair of points and discuss their answers.
4. Ask: "How does slope describe the relationship between two variables?" and introduce the idea that rates and unit rates function similarly.

Activity 2: Brief Review of Slope as Rate of Change

To help students connect slope to real-world rates, they will explore slope as a rate of change in different contexts.

Instructions:

1. Write these real-world scenarios on the board:
   • A car travels 120 miles in 2 hours.
   • A water tank fills at a rate of 5 gallons per minute.
   • A plant grows 3 inches every 4 weeks.
2. Ask students: "How can we express the relationship between time and distance, water, or plant growth?"
3. Guide them to set up ratios for each scenario:
   • 120 miles per 2 hours → $\frac{120}{2} = 60$ miles per hour
   • 5 gallons per 1 minute (already a unit rate)
   • 3 inches per 4 weeks → $\frac{3}{4}$ inches per week
4. Conclude by explaining that rates and unit rates describe changes between quantities just like slope does in a graph. The graph shows the relationship between slope and rate.

Teach

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.

Example 1: Calculating an Hourly Wage Using a Data Table and Graph

Problem: A part-time worker earns different amounts each week depending on the hours worked. The table below shows their earnings for different weeks. What is their hourly wage?

• The x-axis represents hours worked, labeled from 0 to 30.
• The y-axis represents total earnings in dollars, labeled from 0 to 300.
• Plot the points (10,150), (15,225), and (20,300) and draw a straight line through them.

Solution:

1. Observe that the points form a straight line, meaning earnings increase at a constant rate.
2. Use the slope formula: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ Using points (10,150) and (20,300): $$m = \frac{300 - 150}{20 - 10} = \frac{150}{10} = 15$$
3. The slope represents the hourly wage, so the worker earns $15 per hour.

Example 2: Calculating Miles Per Gallon

Problem: A car has traveled 180 miles using 6 gallons of gas. How many miles per gallon does the car get? If the gas tank holds 15 gallons, how far can the car travel on a full tank?

Solution:

1. Calculate miles per gallon: $$\frac{180}{6} = 30$$ The car gets 30 miles per gallon.
2. Find how far the car can travel on a full 15-gallon tank: $$30 \times 15 = 450$$

Final Answer: The car can travel 450 miles on a full tank.

Example 3: Calculating a Unit Rate for Purchasing Food

Problem: A grocery store sells two sizes of orange juice:

• A 64-ounce bottle costs $5.12
• A 32-ounce bottle costs $2.88

Which is the better deal?

Solution:

1. Find the cost per ounce for each size:
2. For the 64-ounce bottle: $$\frac{5.12}{64} = 0.08 \text{ per ounce}$$
3. For the 32-ounce bottle: $$\frac{2.88}{32} = 0.09 \text{ per ounce}$$
4. Since $0.08 per ounce is cheaper than $0.09 per ounce, the 64-ounce bottle is the better deal.

Example 4: Calculating Speed Using a Graph

Problem: A cyclist's journey is represented in the graph below. Find the speed of the cyclist in miles per hour.

Solution:

1. Pick two points from the graph, such as (1,10) and (3,30).
2. Use the formula for speed: $$\text{Speed} = \frac{\text{Distance}}{\text{Time}}$$
3. Calculate using the two points: $$\frac{30 - 10}{3 - 1} = \frac{20}{2} = 10$$
4. The cyclist's speed is 10 miles per hour.

Example 5: Interpreting a Rate from a Graph

Problem: A bakery bakes loaves of bread at a constant rate. The graph below shows the number of loaves baked over time. Find the baking rate in loaves per hour.

Solution:

1. Choose two points from the graph, such as (2,40) and (4,80).
2. Use the rate formula: $$\text{Rate} = \frac{\text{Output}}{\text{Time}}$$
3. Calculate: $$\frac{80 - 40}{4 - 2} = \frac{40}{2} = 20$$
4. The bakery bakes 20 loaves per hour.

Example 6: Interpreting a Rate from a Graph of Water Flow

Problem: A water tank is being filled at a constant rate. The graph below represents the relationship between time and the amount of water in the tank. Find the filling rate in gallons per minute.

Solution:

1. Pick two points from the graph, such as (2,10) and (6,30).
2. Use the rate formula: $$\text{Rate} = \frac{\text{Output}}{\text{Time}}$$
3. Calculate: $$\frac{30 - 10}{6 - 2} = \frac{20}{4} = 5$$
4. The tank is filling at 5 gallons per minute.

Review

Lesson Summary

In this lesson, students explored the concepts of rates and unit rates and their importance in real-world applications. They learned how to:

• Identify and calculate rates by comparing two quantities with different units.
• Determine unit rates by scaling rates down to "per 1" (e.g., miles per hour, cost per item).
• Use graphs to interpret rates as slopes of lines representing proportional relationships.
• Apply rates and unit rates to real-world problems, including wages, fuel efficiency, pricing, and speed.

By understanding these concepts, students are better equipped to analyze proportional relationships and make informed decisions in everyday scenarios.

Key Vocabulary

• Rate: A comparison of two different quantities with different units, such as miles per hour or price per pound.
• Unit Rate: A rate in which the second quantity is 1, such as $2 per apple or 50 miles per hour.
• Proportional Relationship: A relationship between two quantities where the ratio remains constant.
• Slope: The rate of change of a linear function, representing how one quantity changes in relation to another.
• Graph of a Rate: A visual representation of a proportional relationship, often showing a straight line passing through the origin.

Review Activity

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.

Additional Worked-Out Examples

Example 1: Calculating a Rate for Water Usage

Problem: A family uses 600 gallons of water over 4 days. How many gallons of water do they use per day?

Solution:

1. Set up the rate: $$\frac{600 \text{ gallons}}{4 \text{ days}}$$
2. Divide to find the unit rate: $$\frac{600}{4} = 150$$
3. Final Answer: The family uses 150 gallons per day.

Example 2: Finding the Better Buy Using Unit Rates

Problem: A grocery store sells two different packs of cereal:

• Pack A: 18 ounces for $4.50
• Pack B: 24 ounces for $6.00

Which pack is the better deal?

Solution:

1. Find the cost per ounce for each pack.
2. Pack A: $$\frac{4.50}{18} = 0.25 \text{ per ounce}$$
3. Pack B: $$\frac{6.00}{24} = 0.25 \text{ per ounce}$$
4. Since both have the same cost per ounce, either option is a good choice.

Example 3: Speed as a Rate on a Graph

Problem: A high-speed train travels between two cities at a constant speed. The graph below represents its distance over time. Find the train's speed in miles per hour.

Solution:

1. Pick two points from the graph, such as (1,120) and (3,360).
2. Use the slope formula: $$\text{Speed} = \frac{\text{Distance}}{\text{Time}}$$
3. Calculate using the two points: $$\frac{360 - 120}{3 - 1} = \frac{240}{2} = 120$$
4. The train is traveling at 120 miles per hour.

Example 4: Water Flow Rate from a Graph

Problem: A hose fills a swimming pool at a constant rate. The graph below represents the relationship between time and the amount of water in the pool. Find the filling rate in gallons per minute.

Solution:

1. Pick two points from the graph, such as (2,40) and (6,120).
2. Use the rate formula: $$\text{Rate} = \frac{\text{Output}}{\text{Time}}$$
3. Calculate: $$\frac{120 - 40}{6 - 2} = \frac{80}{4} = 20$$
4. The pool is filling at 20 gallons per minute.

Quiz

Answer the following questions.

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?
    
     

Answer Key

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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