SAT Math Lesson Plan 7: Ratios and Proportions

Lesson Summary

This 45-minute lesson focuses on solving problems involving ratios, proportions, and percents—key skills tested in the SAT Math section under the Problem Solving and Data Analysis domain. These concepts appear in approximately 29% of the SAT Math questions. Students will develop fluency in simplifying ratios, solving proportions, converting between fractions, decimals, and percents, and applying percent increase and decrease in real-world contexts. The lesson includes a warm-up activity, five instructional examples, a review section, and a 10-question quiz with solutions. This is part of a comprehensive 35-lesson SAT Math prep library.

Lesson Objectives

Understand and interpret ratios and proportions.
Solve problems involving percents, including percent increase and decrease.
Apply these concepts to real-world scenarios.

Common Core Standards

CCSS.MATH.CONTENT.7.RP.A.1 – Compute unit rates associated with ratios of fractions.
CCSS.MATH.CONTENT.7.RP.A.2 – Recognize and represent proportional relationships between quantities.
CCSS.MATH.CONTENT.7.RP.A.3 – Use proportional relationships to solve multistep ratio and percent problems.

Prerequisite Skills

Basic understanding of fractions and decimals.
Ability to perform arithmetic operations.

Key Vocabulary

Ratio – A comparison of two quantities by division.
Proportion – An equation stating that two ratios are equal.
Percent – A ratio that compares a number to 100.
Unit Rate – A rate with a denominator of one when simplified.

Warm Up

To begin, let’s review the three core concepts of this lesson: ratios, proportions, and percents. These ideas are closely connected and often appear in real-world problems.

Review: Ratios

A ratio is a comparison between two quantities. For example, if a class has 12 boys and 8 girls, the ratio of boys to girls is:

\[ \frac{12}{8} = \frac{3}{2} \]

This means that for every 3 boys, there are 2 girls. Ratios can be simplified like fractions.

Review: Proportions

A proportion is an equation that states two ratios are equal. For example, if a recipe calls for 3 cups of flour for every 2 cups of sugar, and you want to use 9 cups of flour, how much sugar is needed?

Set up a proportion:

\[ \frac{3}{2} = \frac{9}{x} \]

Cross multiply:

\[ 3x = 18 \Rightarrow x = 6 \]

Answer: You’ll need 6 cups of sugar.

Review: Percents

A percent is a special kind of ratio where the denominator is 100. For example, 30% means “30 out of 100.” To find 30% of 80:

\[ 30\% = 0.30, \quad 0.30 \times 80 = 24 \]

Answer: 30% of 80 is 24.

Teach

Ratios, proportions, and percents are among the most common types of questions on the SAT Math section. These concepts fall under the “Problem Solving and Data Analysis” domain, which makes up about 29% of the exam. While the problems might not always use the words “ratio,” “proportion,” or “percent,” the relationships they describe often rely on these concepts. That’s why it’s important to know what to look for.

How Do I Know It’s a Ratio Problem?

Look for questions that compare two quantities or describe parts of a whole. Phrases like “for every,” “per,” or “compared to” are clues that a ratio is involved. Ratios may also be embedded in real-world contexts like recipes, maps, or student-to-teacher ratios.

How Do I Know It’s a Proportion Problem?

If two ratios are set equal, or the problem involves scaling or missing values in a table or double number line, you’re likely working with a proportion. These are often solved using cross multiplication. Questions about prices, distances, or quantities that scale proportionally fall into this category.

How Do I Know It’s a Percent Problem?

Look for the % symbol or words like “percent,” “percentage,” “increase,” or “discount.” SAT percent problems often test real-world applications like tax, tip, percent change (increase or decrease), and percent error. Be prepared to convert between percents, decimals, and fractions.

How Do I Know Which Strategy to Use?

If the problem gives two numbers and asks for a third in the same ratio → set up a proportion.
If you see phrases like “how many times larger” or “what part of” → think ratios and simplify.
If you’re asked to find a percentage of a number → convert the percent to a decimal and multiply.
If the problem involves increase or decrease → find the change and divide by the original amount to get a percent.

In the following examples, we’ll look at all of these scenarios. Take note of the keywords in each problem and how they connect to the strategies being used. Also, look for patterns in the setup—many SAT ratio and percent questions follow predictable formats.

Example 1: Simplifying a Ratio

This is a ratio problem because it compares two quantities—boys and girls in a class. There’s no action or scaling involved, just a relationship.

Problem: A class has 18 boys and 12 girls. What is the simplified ratio of boys to girls?

Step 1: Write the ratio as a fraction: \[ \frac{18}{12} \]

Step 2: Simplify the fraction: \[ \frac{18}{12} = \frac{3}{2} \]

Final Answer: The simplified ratio is 3:2.

Example 2: Solving a Proportion

This is a proportion problem because it gives a relationship (5 pencils cost $2) and asks you to scale it up to 15 pencils. Two ratios are being compared.

Problem: If 5 pencils cost $2, how much do 15 pencils cost?

Step 1: Set up a proportion:

\[ \frac{5}{2} = \frac{15}{x} \]

Step 2: Cross multiply and solve:

\[ 5x = 30 \Rightarrow x = 6 \]

Final Answer: 15 pencils cost $6.

Example 3: Finding a Percent of a Number

This is a percent problem because it asks for “30% of 80.” Problems like this usually involve converting the percent to a decimal and multiplying.

Problem: What is 30% of 80?

Step 1: Convert percent to a decimal: \[ 30\% = 0.30 \]

Step 2: Multiply: \[ 0.30 \times 80 = 24 \]

Final Answer: 30% of 80 is 24.

Example 4: Percent Increase

This is a percent change problem. You’re comparing two prices and looking for how much the price increased as a percent of the original value. These types of questions often appear in sales, population growth, or tax-related scenarios.

General Formula for Percent Increase:

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

Problem: A shirt originally costs $40. It is now priced at $50. What is the percent increase?

Step 1: Identify the original and new values.

Original Value = 40
New Value = 50

Step 2: Use the formula:

\[ \frac{50 - 40}{40} \times 100 = \frac{10}{40} \times 100 = 0.25 \times 100 = 25\% \]

Final Answer: The percent increase is 25%.

Example 5: Real-World Application (Map Scale)

This is a proportion problem. A map scale relates two units (inches and miles), and we’re scaling up a known ratio to find an unknown value.

Problem: A map has a scale of 1 inch = 50 miles. If two cities are 3.5 inches apart on the map, how far apart are they in real life?

Step 1: Set up a proportion:

\[ \frac{1}{50} = \frac{3.5}{x} \]

Step 2: Cross multiply:

\[ x = 3.5 \times 50 = 175 \]

Final Answer: The two cities are 175 miles apart.

Example 6: Percent Decrease

This is a percent change problem, but in reverse. You’re finding how much a value went down compared to its original amount. These problems often appear in sales or depreciation scenarios.

General Formula for Percent Decrease:

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

Problem: A computer originally cost $1,200. It is now on sale for $900. What is the percent decrease?

Step 1: Identify the original and new values.

Original Value = 1,200
New Value = 900

Step 2: Use the formula:

\[ \frac{1200 - 900}{1200} \times 100 = \frac{300}{1200} \times 100 = 0.25 \times 100 = 25\% \]

Final Answer: The percent decrease is 25%.

Example 7: Solving with Percent Equation

This is a percent problem where you're given the final price after a percent decrease, and you have to find the original value. This is common in SAT questions about discounts, markups, or tax calculations where the original price is unknown.

Problem: After a 20% discount, a jacket costs $80. What was the original price?

Step 1: Let x be the original price.

Since it’s a 20% discount, the sale price is 80% of the original price:

\[ 0.80x = 80 \]

Step 2: Solve the equation:

\[ x = \frac{80}{0.80} = 100 \]

Final Answer: The original price was $100.

Review

In this lesson, you explored key strategies for solving SAT Math problems involving ratios, proportions, and percents. These included simplifying ratios, solving proportions using cross multiplication, calculating percent increase and decrease, and solving percent equations. These concepts often appear in real-world scenarios such as prices, maps, recipes, and discounts. In the following examples, you’ll review and apply each of these strategies to reinforce what you’ve learned.

Example 1: Simplifying a Ratio

Problem: A school band has 20 woodwind players and 12 brass players. What is the simplified ratio of woodwinds to brass?

Step 1: Write as a fraction: \[ \frac{20}{12} \]

Step 2: Simplify: \[ \frac{20}{12} = \frac{5}{3} \]

Final Answer: 5:3

Example 2: Percent of a Number

Problem: What is 15% of 240?

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

Step 2: Multiply: \[ 0.15 \times 240 = 36 \]

Final Answer: 36

Example 3: Scale Model (Proportion)

Problem: A scale drawing uses 1 inch = 15 feet. A building measures 4.5 inches on the drawing. How tall is the building?

Step 1: Set up a proportion: \[ \frac{1}{15} = \frac{4.5}{x} \]

Step 2: Cross multiply: \[ x = 4.5 \times 15 = 67.5 \]

Final Answer: The building is 67.5 feet tall.

Example 4: Percent Decrease

Problem: A gym membership that was $80 is now $68. What is the percent decrease?

Step 1: Find the change: \[ 80 - 68 = 12 \]

Step 2: Divide by original: \[ \frac{12}{80} = 0.15 \Rightarrow 15\% \]

Final Answer: 15% decrease

Example 5: Percent Equation (Unknown Original)

Problem: After a 30% discount, a pair of shoes costs $84. What was the original price?

Step 1: Let x be the original price. Since 70% of the price remains after a 30% discount:

\[ 0.70x = 84 \Rightarrow x = \frac{84}{0.70} = 120 \]

Final Answer: The original price was $120.

Self-Study Tip: Practice spotting which type of percent problem you’re working with: Is it a change? A part of a whole? A percent of an unknown? Choose your strategy accordingly.

Multimedia Resources

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

https://www.media4math.com/LessonPlans/SupportResourcesSATMathLesson7

Quiz

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

What is the simplified ratio of 36 to 24?
a) 3:2
b) 2:3
c) 4:3
d) 6:5
If 4 pencils cost \$1.20, what is the cost of 10 pencils?
a) \$2.00
b) \$3.00
c) \$2.50
d) \$3.20
What is 25% of 160?
a) 30
b) 35
c) 40
d) 45
The scale of a map is 1 inch = 20 miles. How many miles apart are two cities that are 6.5 inches apart on the map?
a) 120
b) 110
c) 130
d) 140
A jacket that originally costs \$80 is on sale for \$60. What is the percent decrease?
a) 15%
b) 20%
c) 25%
d) 30%
The ratio of boys to girls in a class is 5:3. If there are 24 students, how many are boys?
a) 10
b) 12
c) 15
d) 18
A bag contains red and blue marbles in a ratio of 3:5. If there are 40 marbles total, how many are red?
a) 15
b) 20
c) 25
d) 30
John earns \$540 for working 36 hours. At the same rate, how much would he earn for 50 hours?
a) \$700
b) \$720
c) \$750
d) \$780
If a 16-ounce bottle of juice costs \$2.40, what is the cost per ounce?
a) \$0.10
b) \$0.12
c) \$0.15
d) \$0.18
The population of a town increased from 18,000 to 22,500. What is the percent increase?
a) 20%
b) 22.5%
c) 25%
d) 30%

Answer Key

Answer: a) 3:2
Solution: Simplify $ \frac{36}{24} = \frac{3}{2} $. The ratio is 3:2.
Answer: b) \$3.00
Solution: Set up a proportion: $ \frac{4}{1.20} = \frac{10}{x} $ Cross multiply: $ 4x = 12 \Rightarrow x = 3 $
Answer: c) 40
Solution: Convert 25% to a decimal: $ 0.25 \times 160 = 40 $
Answer: c) 130
Solution: Use the map scale: $ 6.5 \times 20 = 130 $ miles
Answer: c) 25%
Solution: Decrease: $ 80 - 60 = 20 $ Percent decrease: $ \frac{20}{80} = 0.25 = 25\% $
Answer: c) 15
Solution: Total parts = 5 + 3 = 8 $ \frac{5}{8} \times 24 = 15 $
Answer: a) 15
Solution: Total parts = 3 + 5 = 8 $ \frac{3}{8} \times 40 = 15 $
Answer: b) \$720
Solution: Unit rate: $ \frac{540}{36} = 15 $ $ 15 \times 50 = 750 $ → Correction: That gives \$750, so correct answer is:
Answer: c) $\750
Answer: b) \$0.15
Solution: $ \frac{2.40}{16} = 0.15 $
Answer: c) 25%
Solution: Increase = $ 22,500 - 18,000 = 4,500 $ $ \frac{4,500}{18,000} = 0.25 = 25\% $