# 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.

- Understand and apply ratio concepts to solve complex, multi-step problems
- Use proportional reasoning in various real-world contexts
- Develop problem-solving strategies for challenging ratio-based questions

## TEKS Standards

- 7.4D: Solve problems involving ratios, rates, and percents

## 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.

- Find 20% of 50

Solution: 20% of 50 = 0.20 × 50 = 10 - 15 is what percent of 60?

Solution: (15 ÷ 60) × 100 = 25% - What percent of 80 is 24?

Solution: (24 ÷ 80) × 100 = 30%

If time allows, 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__

## 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:

- 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: 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:

- Calculate the discount: 30% of \$200 = 0.30 * 200 = $60
- Subtract the discount: \$200 - \$60 = \$140 (sale price)
- Calculate the tax: 7% of \$140 = 0.07 * 140 = \$9.80
- 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

- What is 30% of 150?

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

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

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

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

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

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

- What is 12% of 250?

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

- 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

- 45
- $30
- 12 cups
- $300
- $88
- 40 miles per hour
- $57.50
- 30
- $12
- $86.40

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