SAT Math Lesson Plan 8: Percents and Percent Change

Lesson Summary

This 45-minute lesson focuses on understanding, calculating, and applying percents in a variety of contexts. Percents appear frequently on the SAT Math section, particularly within the Problem Solving and Data Analysis domain, which accounts for roughly 29% of the test. Students will review how to convert between fractions, decimals, and percents, solve percent increase and decrease problems, calculate percent error, and work with multi-step percent problems. Instructional examples, review problems, and SAT-style quiz questions ensure students can identify when to apply percent formulas and how to interpret percent language in real-world and abstract contexts. This lesson is part of a 35-lesson SAT Math prep series.

Lesson Objectives

• Convert between fractions, decimals, and percents.
• Use percent formulas to solve problems involving increase, decrease, and error.
• Apply percent concepts to multi-step word problems and real-world situations.

Common Core Standards

• CCSS.MATH.CONTENT.7.RP.A.3 – Use proportional relationships to solve multistep ratio and percent problems.
• CCSS.MATH.CONTENT.6.RP.A.3.C – Find a percent of a quantity as a rate per 100; solve problems involving finding the whole, given a part and the percent.

Prerequisite Skills

• Fluency with basic operations and solving equations.
• Understanding of fractions and decimals.
• Basic familiarity with ratio and proportion concepts.

Key Vocabulary

• Percent – A ratio that compares a number to 100.
• Percent Increase – The amount a value grows, expressed as a percent of the original.
• Percent Decrease – The amount a value shrinks, expressed as a percent of the original.
• Percent Error – A measure of how inaccurate an estimate is, compared to the true value.
• Percent Change – The increase or decrease in a quantity, relative to its original value, expressed as a percent.

Warm Up

To prepare for this lesson, let’s review a few basic percent concepts that are frequently tested on the SAT.

Review 1: Converting Between Forms

• To convert a percent to a decimal: move the decimal point two places to the left.
  Example: \( 35\% = 0.35 \)
• To convert a decimal to a percent: move the decimal point two places to the right.
  Example: \( 0.12 = 12\% \)
• To convert a fraction to a percent: divide and multiply by 100.
  Example: \( \frac{3}{4} = 0.75 = 75\% \)

Review 2: Finding a Percent of a Number

Problem: What is 20% of 150?

Step 1: Convert 20% to decimal: \( 0.20 \)

Step 2: Multiply: \[ 0.20 \times 150 = 30 \]

Answer: 20% of 150 is 30.

Use the Desmos calculator to explore percent problems visually: https://www.desmos.com/scientific

Example 1: Finding a Percent of a Number

This is a basic percent problem. It asks “What is 15% of 200?” which means you simply convert the percent to a decimal and multiply.

Problem: What is 15% of 200?

Step 1: Convert 15% to decimal: \[ 15\% = 0.15 \]

Step 2: Multiply: \[ 0.15 \times 200 = 30 \]

Final Answer: 30

Example 2: Percent Increase

This is a percent change problem involving an increase. These are common on the SAT in sales or data interpretation questions.

Problem: A population grows from 800 to 1,000. What is the percent increase?

Formula for Percent Increase:

\[ \text{Percent Increase} = \frac{\text{New} - \text{Original}}{\text{Original}} \times 100 \]

Step 1: Compute the difference: \[ 1,000 - 800 = 200 \]

Step 2: Apply the formula: \[ \frac{200}{800} \times 100 = 0.25 \times 100 = 25\% \]

Final Answer: 25%

Example 3: Percent Decrease

This is another percent change problem, but this time it involves a decrease. You’ll often see this in SAT word problems about discounts or depreciation.

Problem: A phone originally costs $600 and is on sale for $480. What is the percent decrease?

Formula for Percent Decrease:

\[ \text{Percent Decrease} = \frac{\text{Original} - \text{New}}{\text{Original}} \times 100 \]

Step 1: Compute the decrease: \[ 600 - 480 = 120 \]

Step 2: Apply the formula: \[ \frac{120}{600} \times 100 = 0.20 \times 100 = 20\% \]

Final Answer: 20%

Example 4: Percent Equation (Unknown Original)

This is a reverse percent problem. You’re given the final value after a percent change and asked to find the original value. These questions are often phrased with discount or markup language.

Problem: After a 25% discount, a pair of shoes costs $90. What was the original price?

Step 1: Let x be the original price. Since 75% of the price remains after a 25% discount, we use:

\[ 0.75x = 90 \]

Step 2: Solve for x: \[ x = \frac{90}{0.75} = 120 \]

Final Answer: The original price was $120.

Example 5: Percent Error

This type of problem appears less frequently but is still fair game for the SAT. Percent error measures how far off a guess or measurement is from the actual value.

Formula for Percent Error:

\[ \text{Percent Error} = \frac{|\text{Approximate} - \text{Exact}|}{\text{Exact}} \times 100 \]

Problem: A student estimates that a book has 250 pages, but it actually has 280 pages. What is the percent error?

Step 1: Subtract and take the absolute value: \[ |250 - 280| = 30 \]

Step 2: Divide by actual value: \[ \frac{30}{280} \times 100 \approx 10.7\% \]

Final Answer: The percent error is approximately 10.7%

Review

In this lesson, you learned how to solve a variety of percent problems, including finding a percent of a number, calculating percent increase or decrease, solving reverse percent equations, and estimating percent error. These question types frequently appear on the SAT Math section in both real-world and abstract problem settings. The examples below will help reinforce these skills and prepare you for the quiz.

Example 1: Percent of a Number

Problem: What is 40% of 250?

Step 1: Convert 40% to decimal: \[ 0.40 \]

Step 2: Multiply: \[ 0.40 \times 250 = 100 \]

Final Answer: 100

Example 2: Percent Increase

Problem: A town's population grew from 18,000 to 22,500. What was the percent increase?

Step 1: Calculate the increase: \[ 22,500 - 18,000 = 4,500 \]

Step 2: Divide by the original: \[ \frac{4,500}{18,000} = 0.25 \Rightarrow 25\% \]

Final Answer: 25% increase

Example 3: Percent Decrease

Problem: A car’s value drops from $24,000 to $18,000. What is the percent decrease?

Step 1: Find the difference: \[ 24,000 - 18,000 = 6,000 \]

Step 2: Use the percent decrease formula: \[ \frac{6,000}{24,000} = 0.25 \Rightarrow 25\% \]

Final Answer: 25% decrease

Example 4: Reverse Percent Problem

Problem: After a 20% discount, a laptop costs $720. What was the original price?

Step 1: Let x be the original price. 80% of the price remains after a 20% discount:

\[ 0.80x = 720 \Rightarrow x = \frac{720}{0.80} = 900 \]

Final Answer: The original price was $900.

Example 5: Percent Error

Problem: A lab measures the boiling point of a substance to be 99°C. The actual boiling point is 100°C. What is the percent error?

Step 1: Find the difference: \[ |99 - 100| = 1 \]

Step 2: Divide by the actual value: \[ \frac{1}{100} = 0.01 \Rightarrow 1\% \]

Final Answer: 1% error

Self-Study Tip: On the SAT, percent problems may not always include the % symbol. Look for comparison words like "more than," "less than," or "out of" to identify when percent reasoning is needed.

Multimedia Resources

To explore video tutorials, math examples, and other support resources for this lesson, visit the following page:

https://www.media4math.com/LessonPlans/SupportResourcesSATMathLesson8

Quiz

Directions: Solve each problem. Choose the best answer from the choices provided. Show your work on a separate sheet of paper.

1. What is 12% of 150?
   a) 18
   b) 20
   c) 22
   d) 24
2. After a 30% increase, the price of a TV is $650. What was the original price?
   a) $455
   b) $500
   c) $520
   d) $550
3. A pair of shoes is marked down 25% from its original price of $160. What is the sale price?
   a) $120
   b) $125
   c) $130
   d) $140
4. What is the percent increase from 40 to 50?
   a) 20%
   b) 22%
   c) 25%
   d) 30%
5. If 18 is 60% of what number?
   a) 24
   b) 27
   c) 30
   d) 36
6. A student estimated that a book had 280 pages, but it actually had 300. What is the percent error?
   a) 5%
   b) 6.5%
   c) 6.67%
   d) 7%
7. A population decreases from 20,000 to 18,000. What is the percent decrease?
   a) 8%
   b) 9%
   c) 10%
   d) 12%
8. Which expression represents a 15% increase on price x?
   a) 0.85x
   b) 1.15x
   c) x + 0.15
   d) x ÷ 0.85
9. After a 40% decrease, a jacket costs $90. What was the original price?
   a) $120
   b) $130
   c) $140
   d) $150
10. 80 is what percent of 200?
    a) 30%
    b) 35%
    c) 40%
    d) 45%

Answer Key

1. Answer: a) 18
   \( 0.12 \times 150 = 18 \)
2. Answer: b) $500
   Let x be the original price. \( 1.30x = 650 \Rightarrow x = \frac{650}{1.3} = 500 \)
3. Answer: a) $120
   \( 0.25 \times 160 = 40 \Rightarrow 160 - 40 = 120 \)
4. Answer: c) 25%
   \( \frac{50 - 40}{40} = \frac{10}{40} = 0.25 = 25\% \)
5. Answer: c) 30
   Let x be the unknown number. \( 0.60x = 18 \Rightarrow x = \frac{18}{0.60} = 30 \)
6. Answer: c) 6.67%
   \( \frac{|280 - 300|}{300} = \frac{20}{300} = 0.0667 = 6.67\% \)
7. Answer: c) 10%
   \( \frac{20000 - 18000}{20000} = \frac{2000}{20000} = 0.10 = 10\% \)
8. Answer: b) 1.15x
   A 15% increase is the same as multiplying by \( 1 + 0.15 = 1.15 \)
9. Answer: d) $150
   Let x be the original price. \( 0.60x = 90 \Rightarrow x = \frac{90}{0.60} = 150 \)
10. Answer: c) 40%
    \( \frac{80}{200} = 0.40 = 40\% \)