# Lesson Plan: Solving Percent Problems and Using Proportional Reasoning

This lesson plan is designed for two 50-minute class periods.

## Lesson Objectives

• Solve various types of percent problems
• Use proportional relationships in percent calculations
• Apply percent concepts to financial situations
• Compare simple and compound interest

## TEKS Standards

• 8.4D: Use proportional relationships to solve multi-step ratio and percent problems.
• 8.5A: Represent linear proportional situations with tables, graphs, and equations.
• 8.5E: Solve problems using direct variation.
• 8.12D: Calculate and compare simple interest and compound interest earnings.

## Prerequisite Skills

• Understanding of percents
• Basic algebra skills
• Knowledge of square roots and cube roots
• Understanding of ratios and proportions

## Key Vocabulary

• Percent
• Markup
• Markdown
• Commission
• Interest
• Irrational number
• Square root
• Cube root
• Rational approximation
• Scale model

## Warm-up Activity (10 minutes)

Review strategies for the following percent calculations:

## Teach (70 minutes)

### Definitions

• Percent: A ratio that compares a number to 100
• Markup: An increase in the price of a product
• Markdown: A decrease in the price of a product
• Commission: A fee paid to an agent or employee for conducting a transaction
• Interest: Money paid regularly at a particular rate for the use of borrowed money
• Irrational number: A number that cannot be expressed as a simple fraction
• Square root: A value that, when multiplied by itself, gives the number
• Cube root: A value that, when multiplied by itself twice, gives the number
• Rational approximation: An estimate of an irrational number using a ratio of integers
• Scale model: A representation of an object that is larger or smaller than the actual size

You can also use this slide show of definitions, which include examples of the relevant term:

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

### Instruction

Demonstrate how to set up proportions to solve problems. Use this slide show to review examples of solving different proportions:

https://www.media4math.com/library/slideshow/math-examples-solving-proportions-algebraically

Use this slide show to give an overview of percents:

https://www.media4math.com/library/slideshow/overview-percents

Next, review these examples.

### Example 1: Markup Problem

A store buys a jacket for $80 and wants to mark it up by 45%. What should the selling price be? Solution:  1. Set up the proportion: 45/100 = x/80 2. Cross multiply: 45 * 80 = 100x 3. Solve for x: x = (45 * 80) / 100 = 36 4. The markup amount is \$36 5. Add the markup to the original price: \$80 + \$36 = \$116 The selling price should be \$116.

### Example 2: Commission Problem

A real estate agent earns a 6% commission on home sales. If they sell a house for \$250,000, how much commission will they earn? Solution:  1. Set up the proportion: 6/100 = x/250,000 2. Cross multiply: 6 * 250,000 = 100x 3. Solve for x: x = (6 * 250,000) / 100 = 15,000 The agent will earn \$15,000 in commission.

### Example 3: Approximating Irrational Numbers

Approximate √20 to the nearest tenth and express it as a ratio.

Solution:

 1. Find the perfect squares on either side of 20: 16 (42) and 25 (52) 2. √20 is between 4 and 5 3. Use a calculator to find √20 ≈ 4.472135... 4. Round to the nearest tenth: √20 ≈ 4.5 5. Express as a ratio: 45:10 or 9:2

### Example 4: Scale Model Problem

An architect is creating a scale model of a building. The actual building is 45 meters tall, and in the model, it is 15 centimeters tall. If a window on the model is 2 centimeters tall, how tall is the actual window?

Solution:

 1. Set up the proportion: 15 cm / 45 m = 2 cm / x m 2. Convert 45 m to cm: 45 m = 4500 cm 3. Rewrite the proportion: 15 / 4500 = 2 / x 4. Cross multiply: 15x = 2 * 4500 5. Solve for x: x = (2 * 4500) / 15 = 600 6. Convert 600 cm to meters: 600 cm = 6 m

The actual window is 6 meters tall.

### Example 5: Carbon Dating

Use this slide show to introduce an application of proportional reasoning in the context of carbon dating:

https://www.media4math.com/library/slideshow/applications-proportional-reasoning-carbon-dating

This table summarizes the data:

 C-14 C-12 Age 1 1.00 • 1012 -- 1 5.00 • 1011 5730 1 2.50 • 1011 11,460 1 1.25 • 1011 17,190 1 6.25 • 1010 22,920 1 3.13 • 1010 28,650 1 1.56 • 1010 34,380 1 7.81 • 109 40,110 1 3.91 • 109 51,570

Make a note of the changing ratios. With each subsequent ratio, the number in scientific notation is reduced by 50% and the age of the artefact is an additional 5730 years old.

## Review (30 minutes)

Refer to the following videos to review key concepts:

## Quiz Questions

1. A store buys a television for $400 and marks it up by 35%. What is the selling price? 2. A real estate agent earns a 5.5% commission on a house sale. How much will they earn if the house sells for$280,000?

3. Approximate √18 to the nearest tenth and express your answer as a ratio.

4. An architect's scale model has a 1:150 ratio. If a door in the model is 3 cm tall, how tall is the actual door?

5. A shirt originally priced at $50 is on sale for 20% off. What is the sale price? 6. If the population of a city increased from 50,000 to 57,500, what was the percent increase? 7. A car's value depreciates from$25,000 to $21,250 after one year. What is the percent decrease? 8. Estimate √8 by finding two perfect squares it falls between, then narrow it down to a range of tenths. 9. A salesperson earns an 8% commission. How much will they earn on a$1500 sale?

10. In a scale model, 2 cm represents 5 m. How many centimeters would represent 12.5 m?

11. Approximate π to two decimal places and express your answer as a ratio.

12. Which is greater: √13 or 3.7? Justify your answer using rational approximations.

1. $540 2.$15,400
5. $40 6. 15% 7. -15% 8. 2.8 < √8 < 2.9 (between 2^2=4 and 3^2=9) 9.$120