Lesson Plan: Solving Multi-step Ratio and Percent Problems

Lesson Summary

In this lesson, students will learn how to solve multi-step ratio and proportion problems. These problems often involve multiple operations, unit conversions, or comparisons between different ratios. The lesson will reinforce strategies for breaking down complex ratio problems into manageable steps and using proportional reasoning to find solutions.

Students will practice:

• Setting up and solving multi-step ratio equations.
• Using proportions to solve real-world problems.
• Applying unit conversions within ratio problems.
• Interpreting and analyzing word problems involving ratios and rates.

By the end of this lesson, students will have a deeper understanding of how ratios and proportions are applied in everyday situations, including recipes, speed calculations, and scale models.

Lesson Objectives

• Solve multi-step ratio problems
• Calculate percentages in various contexts
• Use proportional relationships to solve percent problems

Common Core Standards

• 7.RP.A.3 Use proportional relationships to solve multistep ratio and percent problems.

Prerequisite Skills

• Understanding of ratios and proportions
• Basic percentage calculations

Key Vocabulary

• Ratio: A comparison of two numbers, often expressed as a fraction $\frac{a}{b}$ or using a colon (e.g., 3:5).
• Proportion: An equation stating that two ratios are equal, such as $\frac{3}{4} = \frac{6}{8}$.
• Unit Rate: A rate in which the denominator is 1, such as $\frac{60}{1}$ miles per hour.
• Constant of Proportionality: The fixed ratio between two proportional quantities, represented by $k$ in the equation $y = kx$.
• Scaling: The process of increasing or decreasing a ratio while maintaining its proportional relationship.
• Unit Conversion: Changing measurements from one unit to another while maintaining the ratio (e.g., converting inches to feet).
• Cross Multiplication: A method used to solve proportions by multiplying diagonally and solving for the unknown value.

Multimedia Resources

• Intro to Ratios and Unit Rates
• Unit Rates and Complex Fractions
• Proportional Relationships
• Representing Proportional Relationships

Warm Up Activities

Choose from one or more of these activities.

Activity 1: Simple Percents

Calculate simple percentages: Provide students with three types of percentage problems to solve.

1. Find 20% of 50
   Solution: 20% of 50 = 0.20 × 50 = 10
2. 15 is what percent of 60?
   Solution: (15 ÷ 60) × 100 = 25%
3. What percent of 80 is 24?
   Solution: (24 ÷ 80) × 100 = 30%

Activity 2: Strategies for Calculating Percents

Review these slide show, which provide strategies for calculating percents:

• Percent of a number: https://www.media4math.com/library/64365/asset-preview
• Percent one number is of another: https://www.media4math.com/library/64368/asset-preview

Activity 3: Percent-Fraction-Decimal Equivalents

Understanding the relationship between percents, fractions, and decimals is essential for solving ratio problems. Start by reviewing common conversions.

Common Equivalents:

• $25\% = \frac{1}{4} = 0.25$
• $50\% = \frac{1}{2} = 0.5$
• $75\% = \frac{3}{4} = 0.75$
• $10\% = \frac{1}{10} = 0.1$
• $20\% = \frac{1}{5} = 0.2$

Example: Convert 40% to a fraction and a decimal.

Solution:

1. Write 40% as a fraction: $$\frac{40}{100}$$
2. Simplify: $$\frac{40}{100} = \frac{2}{5}$$
3. Convert to a decimal by dividing 40 by 100: $$0.4$$

Teach

Definitions

• Percentage: A proportion or share in relation to a whole, expressed as a number out of 100
• Markup: The amount added to the cost price of goods to cover overhead and profit
• Discount: A reduction from the usual or list price
• Commission: A fee paid to an agent or employee for transacting a piece of business or performing a service
• Tax: A compulsory contribution to state revenue, levied by the government on workers' income and business profits, or added to the cost of some goods, services, and transactions
• Tip: A sum of money given voluntarily or beyond obligation usually for some service

Use this slide show, which includes definitions for these and other related terms:

https://www.media4math.com/library/slideshow/definitions-percent-calculations

Instruction

Introduce these videos, which cover various topics in proportions and percent calculations:

• Proportions: https://www.media4math.com/library/1798/asset-preview
• Solving Proportions: https://www.media4math.com/library/1799/asset-preview
• Calculating tips and commissions: https://www.media4math.com/library/1819/asset-preview
• Calculating tax: https://www.media4math.com/library/1818/asset-preview
• Percent increase: https://www.media4math.com/library/1815/asset-preview
• Percent decrease: https://www.media4math.com/library/1816/asset-preview

Example 1: Recipe Scaling

A recipe calls for $\frac{2}{3}$ cup of sugar to make 4 servings. How much sugar is needed for 10 servings?

Solution:

1. Set up a proportion where $x$ represents the amount of sugar needed for 10 servings: $$\frac{2}{3} \div 4 = \frac{x}{10}$$
2. Find the unit rate by dividing $\frac{2}{3}$ by 4: $$\frac{2}{3} \times \frac{1}{4} = \frac{2}{12} = \frac{1}{6}$$
3. Multiply by 10 to find the total sugar needed: $$x = \frac{1}{6} \times 10 = \frac{10}{6} = \frac{5}{3}$$

Thus, $\frac{5}{3}$ cups of sugar are needed for 10 servings.

Example 2: Speed and Distance

A car travels 150 miles in 3 hours. How far will it travel in 8 hours at the same speed?

Solution:

1. Find the unit rate (miles per hour): $$\frac{150}{3} = 50 \text{ miles per hour}$$
2. Multiply by 8 to find the total distance: $$50 \times 8 = 400$$

Thus, the car will travel 400 miles in 8 hours.

Example 3: Currency Exchange

A traveler exchanges \$200 for euros at a rate of 1 USD = 0.85 EUR. How much will they receive in euros?

Solution:

1. Set up the proportion where $x$ represents the amount in euros: $$\frac{1}{0.85} = \frac{200}{x}$$
2. Use cross-multiplication: $$1 \times x = 200 \times 0.85$$
3. Simplify: $$x = 170$$

Thus, the traveler will receive 170 euros.

Example 4: Scale Drawings

A blueprint of a building uses a scale of 1 inch = 5 feet. If the height of the building in the blueprint is 12 inches, what is the actual height?

Solution:

1. Set up the proportion where $x$ represents the actual height: $$\frac{1}{5} = \frac{12}{x}$$
2. Use cross-multiplication: $$1 \times x = 5 \times 12$$
3. Simplify: $$x = 60$$

Thus, the actual height of the building is 60 feet.

Example 5: Discount on a Jacket

A jacket originally costs \$80 and is on sale for 25% off. After the discount, a 7% sales tax is applied. What is the final price of the jacket?

Solution:

1. Find the discount amount: $$80 \times 0.25 = 20$$
2. Subtract the discount from the original price: $$80 - 20 = 60$$
3. Calculate the sales tax: $$60 \times 0.07 = 4.20$$
4. Add the tax to the discounted price: $$60 + 4.20 = 64.20$$

Thus, the final price of the jacket is \$64.20.

Example 6: Commission on a Sale

A salesperson earns a 12% commission on every sale. If they sell a \$1,500 laptop and a \$2,000 TV, how much total commission do they earn?

Solution:

1. Calculate the commission for the laptop: $$1500 \times 0.12 = 180$$
2. Calculate the commission for the TV: $$2000 \times 0.12 = 240$$
3. Find the total commission: $$180 + 240 = 420$$

Thus, the salesperson earns a total commission of \$420.

Review

Lesson Summary

In this lesson, students learned how to solve multi-step ratio and proportion problems using different mathematical strategies. They explored real-world applications of ratios, including unit conversions, scaling, and proportional reasoning. The lesson emphasized breaking down complex problems into manageable steps and applying algebraic methods such as cross-multiplication.

Key takeaways from the lesson:

• Multi-step ratio problems require careful analysis and sometimes involve unit conversions or multiple proportions.
• Proportional relationships can be represented using tables, graphs, and equations in the form $y = kx$.
• Cross-multiplication is an effective technique for solving proportion equations.
• Percent-fraction-decimal conversions are essential for solving real-world ratio problems.
• Ratios and proportions appear in practical situations such as scale drawings, financial calculations, and speed/distance problems.

Key Vocabulary

• Ratio: A comparison of two numbers, often expressed as $\frac{a}{b}$ or using a colon (e.g., 3:5).
• Proportion: An equation that states two ratios are equal, such as $\frac{3}{4} = \frac{6}{8}$.
• Constant of Proportionality: The fixed ratio between two proportional quantities, represented by $k$ in $y = kx$.
• Unit Rate: A rate in which the denominator is 1, such as $\frac{60}{1}$ miles per hour.
• Scaling: The process of increasing or decreasing a ratio while maintaining its proportional relationship.
• Unit Conversion: Changing measurements from one unit to another while maintaining the ratio (e.g., converting inches to feet).
• Cross Multiplication: A method used to solve proportions by multiplying diagonally and solving for the unknown value.
• Percent-Fraction-Decimal Conversion: The process of changing a percentage into a fraction or decimal (e.g., $25\% = \frac{1}{4} = 0.25$).

By reviewing these concepts, students will reinforce their understanding of multi-step ratio problems and their real-world applications.

Practice solving multi-step ratio and percent problems: Provide students with practice problems that require them to apply what they have learned.

Example 1: Running Pace

A runner completes 3 miles in 24 minutes. At the same pace, how long will it take to run 10 miles?

Solution:

1. Find the unit rate (minutes per mile): $$\frac{24}{3} = 8 \text{ minutes per mile}$$
2. Multiply by 10 to find the total time: $$8 \times 10 = 80$$

Thus, it will take 80 minutes to run 10 miles at the same pace.

Example 2: Paint Mixing

A painter mixes 2 cups of blue paint with 5 cups of white paint to create a shade of light blue. How many cups of white paint are needed if 6 cups of blue paint are used?

Solution:

1. Set up the proportion: $$\frac{2}{5} = \frac{6}{x}$$
2. Use cross-multiplication: $$2 \times x = 5 \times 6$$
3. Simplify: $$2x = 30$$
4. Divide by 2: $$x = \frac{30}{2} = 15$$

Thus, 15 cups of white paint are needed.

Example 3: Movie Ticket Pricing

Three movie tickets cost \$27. How much will 7 tickets cost at the same rate?

Solution:

1. Find the unit price per ticket: $$\frac{27}{3} = 9 \text{ dollars per ticket}$$
2. Multiply by 7 to find the total cost: $$9 \times 7 = 63$$

Thus, 7 tickets will cost \$63.

Example 4: Water Tank Filling

A water tank fills at a rate of 15 gallons in 4 minutes. How much water will fill in 10 minutes at the same rate?

Solution:

1. Set up the proportion: $$\frac{15}{4} = \frac{x}{10}$$
2. Use cross-multiplication: $$15 \times 10 = 4 \times x$$
3. Simplify: $$150 = 4x$$
4. Divide by 4: $$x = \frac{150}{4} = 37.5$$

Thus, 37.5 gallons of water will fill the tank in 10 minutes.

Example 5: Investment Growth

An investment of \$5,000 earns 8% interest per year. How much will the investment be worth after 2 years, assuming interest is applied annually?

Solution:

1. Find the interest earned in the first year: $$5000 \times 0.08 = 400$$
2. Calculate the new total after the first year: $$5000 + 400 = 5400$$
3. Find the interest earned in the second year: $$5400 \times 0.08 = 432$$
4. Calculate the total after the second year: $$5400 + 432 = 5832$$

Thus, the investment will be worth \$5,832 after 2 years.

Example 6: Price Increase on Electronics

A gaming console originally costs \$400, but the price increases by 15%. After the price increase, a 5% luxury tax is applied. What is the final price?

Solution:

1. Find the price increase: $$400 \times 0.15 = 60$$
2. Add the increase to the original price: $$400 + 60 = 460$$
3. Calculate the luxury tax: $$460 \times 0.05 = 23$$
4. Find the final price: $$460 + 23 = 483$$

Quiz

Answer the following questions.

1. A laptop originally costs \$1,200 and is on sale for 20% off. After the discount, a 6% sales tax is applied. What is the final price of the laptop?
   
    
2. A real estate agent earns a 5% commission on property sales. If they sell one house for \$250,000 and another for \$320,000, how much total commission do they earn?
   
    
3. An investment of \$3,000 earns 7% interest per year. How much will the investment be worth after 2 years, assuming interest is applied annually?
   
    
4. A worker's salary is \$50,000 per year. They receive a 10% raise, but after the raise, 12% of their new salary is deducted for taxes. What is their final salary after taxes?
   
    
5. A recipe requires $\frac{3}{4}$ cup of milk for 4 servings. How much milk is needed for 10 servings?
   
    
6. A television is originally priced at \$900. The store offers a 15% discount and then applies a 9% sales tax to the discounted price. What is the final cost of the television?
   
    
7. A car travels 180 miles on 6 gallons of gas. How far can it travel on 15 gallons at the same rate?
   
    
8. A student correctly answers 42 questions on a 60-question test. What percentage of questions did they answer correctly?
   
    
9. A store sells apples at \$3 per pound. If a customer buys 7.5 pounds, how much do they pay?
   
    
10. A sweater is marked up 25% from its wholesale price of \$48. After the markup, the store has a 10% off sale. What is the final price of the sweater?

Answer Key

1. \$1,017.60
2. \$28,500
3. \$3,434.70
4. \$48,400
5. $\frac{15}{8}$ cups (or 1.875 cups)
6. \$832.95
7. 450 miles
8. 70%
9. \$22.50
10. \$52.80

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