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 Lesson Plan: Solving Multi-step Ratio and Percent Problems

Lesson Objectives

Note: This lesson can be completed in one 50-minute class period. If more practice or review is needed, it can be extended to two class periods.

  • Review ratio concepts and proportional relationships
  • Identify and set up ratios in complex scenarios
  • Apply ratio reasoning to solve multi-step problems
  • Use various strategies such as unit rates, equivalent ratios, and cross multiplication
  • Interpret and communicate solutions in context
  • Apply ratio concepts to solve percentage problems
  • Find the whole given a part and the percent
  • Solve multi-step real-world problems involving percentages using various methods

Florida BEST Standards

  • MA.7.AR.3.1: Apply previous understanding of percentages and ratios to solve multi-step real-world problems.
  • MA.7.AR.3.2: Apply previous understanding of ratios to solve real-world problems involving percentages. Find the whole given a part and the percent.
  • MA.7.AR.3.3: Solve multi-step real-world problems involving percentages using any method.

Prerequisite Skills

  • Understanding of ratios and proportions
  • Basic percentage calculations

Key Vocabulary

  • Percentage
  • Markup
  • Discount
  • Commission
  • Tax
  • Tip

Warm-up Activity (10 minutes)

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%

If time allows, review these slide show, which provide strategies for calculating percents:

Teach (25 minutes)

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:

Example 1: Scaling a Recipe

Demonstrate solving multi-step ratio problems using a recipe scenario.

If a recipe requires 3 cups of flour for every 2 cups of sugar, how much flour is needed for 5 cups of sugar?

Solution: Set up the proportion and solve:

 3/2 = x/5

Cross-multiply:

3 * 5 = 2 * x

15 = 2x

Solving for x gives 

x = 7.5 cups of flour

Example 2: Percentage Discount

Explain how to use proportional relationships in percent problems using a store discount scenario.

A store is having a 25% off sale on all items. If an item originally costs \$80, what is the sale price?

Solution: 

  • Calculate 25% of \$80, which is 0.25 * 80 = \$20. 
  • Subtract this from the original price: \$80 - \$20 = \$60.

Example 3: Commission Calculation

Show strategies for solving complex percentage situations using a sales commission scenario.

A salesperson earns a 5% commission on sales. If they sell \$2000 worth of products, how much commission do they earn?

Solution: Calculate 5% of \$2000, which is 0.05 * 2000 = \$100.

Example 4: Discount and Tax Calculation

Demonstrate how to calculate the price after a discount and then apply sales tax.

An item originally costs \$200. It's on sale for 30% off, and there's a 7% sales tax. What's the final price?

Solution:

  1. Calculate the discount: 30% of \$200 = 0.30 * 200 = $60
  2. Subtract the discount: \$200 - \$60 = \$140 (sale price)
  3. Calculate the tax: 7% of \$140 = 0.07 * 140 = \$9.80
  4. Add the tax to the sale price: \$140 + \$9.80 = \$149.80

The final price is \$149.80.

Review (10 minutes)

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

Example 1: Percent Decrease

Calculate the percent decrease in a product's price.

A laptop was originally priced at \$800. It is now on sale for \$680. What is the percent decrease?

Solution: Calculate the difference: \$800 - \$680 = \$120
Find the percent decrease: (120 / 800) * 100 = 15%
The laptop's price has decreased by 15%.

Example 2: Percent Increase

Calculate the percent increase in a town's population.

A town's population was 25,000 last year. This year, it has grown to 27,500. What is the percent increase?

Solution: Calculate the difference: 27,500 - 25,000 = 2,500
Find the percent increase: (2,500 / 25,000) * 100 = 10%
The town's population has increased by 10%.

Assess (5 minutes)

10-question quiz: Distribute a quiz to assess students' understanding of multi-step ratio and percent problems.

Quiz

  1. What is 30% of 150?

     
  2. If a shirt costs \$40 and is on sale for 25% off, what is the sale price?

     
  3. A recipe calls for 4 cups of water for every 3 cups of rice. How much water is needed for 9 cups of rice?

     
  4. A salesperson earns a 6% commission on sales. If they sell \$5000 worth of products, how much commission do they earn?

     
  5. Calculate the final price of an item that lists for \$100 after a 20% discount and a 10% tax.

     
  6. If a car travels 60 miles in 1.5 hours, what is the speed in miles per hour?

     
  7. A store marks up the price of an item by 15%. If the original price is \$50, what is the new price?

     
  8. What is 12% of 250?

     
  9. If a meal costs \$80 and you want to leave a 15% tip, how much is the tip?

     
  10. A jacket is originally priced at $120. It is first marked down by 20%, and then an additional 10% off the reduced price. What is the final price?

Answer Key

  1. 45
  2. $30
  3. 12 cups
  4. $300
  5. $88
  6. 40 miles per hour
  7. $57.50
  8. 30
  9. $12
  10. $86.40

 

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