Lesson Plan: Word Problems and Real-World Applications

Lesson Summary

This 45-minute SAT Math lesson focuses on strategies for interpreting and solving real-world word problems, which account for an estimated 15–20% of SAT Math questions. These problems require translating written scenarios into mathematical models using algebra, geometry, and statistics. Students will practice reading carefully, identifying relationships, and solving multi-step problems that require estimation, modeling, and interpretation. Word problems span topics such as unit conversions, proportions, graph analysis, and spatial reasoning. This lesson is part of the Problem Solving and Data Analysis domain, which makes up approximately 29% of the SAT Math section. Students will engage with instructional examples, review problems, and a 10-question quiz with detailed solutions. This lesson is part of a 35-lesson SAT Math prep series.

Lesson Objectives

• Analyze and break down word problems into mathematical models.
• Solve multi-step problems using algebra, geometry, and statistics.
• Use problem-solving strategies such as visualization, estimation, and checking solutions.
• Apply mathematical reasoning to real-world scenarios.

Common Core Standards

• CCSS.MATH.CONTENT.HSA.CED.A.1 – Create equations and inequalities in one variable and use them to solve problems.
• CCSS.MATH.CONTENT.7.RP.A.3 – Use proportional relationships to solve multistep ratio and percent problems.
• CCSS.MATH.CONTENT.HSS.ID.A.3 – Interpret differences in shape, center, and spread in the context of the data.

Prerequisite Skills

• Understanding of basic algebra, geometry, and statistics.
• Ability to set up and solve equations.
• Familiarity with unit conversions and proportional reasoning.

Key Vocabulary

• Mathematical Modeling – Translating real-world situations into mathematical expressions or equations.
• Multi-Step Problem – A problem requiring multiple calculations or steps to reach a solution.
• Estimation – Approximating a value to check the reasonableness of an answer.
• Statistical Analysis – Using measures of central tendency (mean, median, mode) to interpret data.
• Proportional Reasoning – Using ratios and proportions to compare values.

Warm Up

Let’s begin with two basic word problems to help activate your problem-solving mindset. These examples involve translating real-world language into mathematical operations.

Warm-Up Problem 1: Proportional Reasoning

Problem: A recipe that makes 4 servings calls for 2 cups of flour. How much flour is needed to make 10 servings?

Step 1: Set up a proportion: $$\frac{2 \, \text{cups}}{4 \, \text{servings}} = \frac{x}{10 \, \text{servings}}$$

Step 2: Cross-multiply: $$4x = 20 \Rightarrow x = 5$$

Answer: 5 cups of flour

Warm-Up Problem 2: Percent Application

Problem: An item originally costs \$80. If it is discounted by 25%, what is the sale price?

Step 1: Find 25% of 80: $$0.25 \times 80 = 20$$

Step 2: Subtract the discount: $$80 - 20 = 60$$

Answer: \$60

For visualizing and checking your solutions, you can use the Desmos calculator: https://www.desmos.com/scientific

Teach

On the SAT, word problems require you to extract key information from written scenarios and translate it into a solvable math model. These problems may involve multiple steps, unit conversions, or combining concepts like percentages, ratios, or statistics. The key is identifying the relationships and operations required to solve the problem.

How to Approach Word Problems

• Read carefully: Identify quantities, units, and what the question is asking.
• Highlight relationships: Look for words like “per,” “total,” “difference,” or “increase.”
• Assign variables: For unknowns or quantities that change, let variables represent them.
• Write an equation or model: Convert the scenario into an equation or inequality.
• Solve and check: Do the math, and then plug your solution back into the context to check for reasonableness.

In the examples that follow, you’ll encounter common SAT-style word problems involving algebra, proportions, percent change, and real-world data. Each example provides detailed reasoning and step-by-step solutions to help you build confidence and fluency.

Example 1: Algebraic Word Problem

This is a classic SAT-style word problem that involves translating a real-world scenario into an algebraic equation.

Problem: A movie theater charges \$10 for each adult ticket and \$6 for each child ticket. A group of 8 people attends a movie. If there are three times as many children as adults and the group spends \$56 total, how many children are in the group?

Step 1:

Let x be the number of adults. Then the number of children is $3x$.

Step 2:

Set up an equation for the total number of people: $$x + 3x = 4x = 8 \Rightarrow x = 2$$

Step 3:

So, there are 2 adults and $3 \times 2 = 6$ children.

Step 4:

Now calculate the total cost: $$2 \times 10 = 20 \quad \text{(adults)}$$ $$6 \times 6 = 36 \quad \text{(children)}$$ $$20 + 36 = 56$$

Final Answer: 6 children and 2 adults

Example 2: Ratio Problem

This problem involves part-to-part and part-to-whole ratio understanding—common in data-based SAT problems.

Problem: In a class, the ratio of boys to girls is 3:5. If there are 24 students total, how many are boys?

Step 1: Add parts of the ratio: $$3 + 5 = 8 \text{ total parts}$$

Step 2: Divide total students by 8: $$\frac{24}{8} = 3$$

Step 3: Multiply by the number of ratio of boys: $$3 \times 3 = 9$$

Answer: There are 9 boys in the class.

Example 3: Percent Increase

This is a percent change word problem. SAT questions often ask you to interpret or calculate percentage increases or decreases.

Problem: The price of a textbook increased from \$60 to \$75. What is the percent increase?

Step 1: Use the Percent Increase Formula

$$\text{Percent Increase} = \left( \frac{\text{New} - \text{Original}}{\text{Original}} \right) \times 100$$

Step 2: Plug in the values

$$\text{Percent Increase} = \left( \frac{75 - 60}{60} \right) \times 100 = \left( \frac{15}{60} \right) \times 100$$

Step 3: Simplify

$$\frac{15}{60} = 0.25 \Rightarrow 0.25 \times 100 = 25\%$$

Final Answer: 25% increase

Example 4: Real-World Data Application

This problem models a real-world situation using a proportion. SAT often uses maps, blueprints, or infographics for such problems.

Problem: On a map, 1 inch represents 20 miles. If the distance between two cities is 3.5 inches on the map, what is the actual distance?

Step 1:

Set up a proportion using the scale. $$\frac{1 \, \text{inch}}{20 \, \text{miles}} = \frac{3.5 \, \text{inches}}{x \, \text{miles}}$$

Step 2:

Cross-multiply and solve: $$1 \cdot x = 3.5 \cdot 20 = 70 \Rightarrow x = 70$$

Final Answer: The actual distance between the two cities is 70 miles.

Example 5: Real-World Work Problem

This is a classic SAT work problem involving time and combined rates. These problems are best approached by expressing work as a rate (jobs per unit of time).

Problem: Machine A can complete a task in 6 hours. Machine B can do the same task in 4 hours. How long will it take both machines working together to complete the task?

Step 1: Express each machine’s rate as a job per hour

• Machine A: $\frac{1 \, \text{job}}{6 \, \text{hours}}$
• Machine B: $\frac{1 \, \text{job}}{4 \, \text{hours}}$

Step 2: Add the two rates together

$$\frac{1}{6} + \frac{1}{4} = \frac{2}{12} + \frac{3}{12} = \frac{5}{12} \, \text{jobs per hour}$$

This means that together, the two machines complete $\frac{5}{12}$ of the job in one hour.

Step 3: Use the combined rate to find the total time

$$\text{Time} = \frac{1 \, \text{job}}{\frac{5}{12} \, \text{jobs/hour}} = \frac{12}{5} = 2.4 \, \text{hours}$$

Step 4: Convert decimal to hours and minutes

$$0.4 \, \text{hours} = 0.4 \times 60 = 24 \, \text{minutes}$$

Final Answer: 2 hours and 24 minutes

Review

In this lesson, you developed strategies for translating real-world scenarios into mathematical models, solving problems using equations, proportions, and percent reasoning. These skills are essential for tackling multi-step word problems on the SAT Math section. Many SAT questions don’t immediately present numbers to calculate—instead, they test how well you can interpret and organize information to arrive at the correct solution.

Here's a recap of the problem-solving tools you practiced:

• Identifying key values and relationships from word problems
• Translating verbal descriptions into algebraic expressions or equations
• Using ratios and proportions to model real-world contexts
• Calculating percent change and interpreting data
• Solving multi-step problems using logical reasoning

Example 1: Linear Equation from a Word Problem

Problem: A gym charges a \$25 monthly fee plus \$5 per class. If a person pays \$70 in one month, how many classes did they attend?

Step 1: Let x represent the number of classes. $$25 + 5x = 70$$

Step 2: Solve: $$5x = 45 \Rightarrow x = 9$$

Answer: 9 classes

Example 2: Real-World Proportions

Problem: A car travels 180 miles on 6 gallons of gas. How far can it go on 10 gallons at the same rate?

Step 1: Set up a proportion: $$\frac{180}{6} = \frac{x}{10}$$

Step 2: Solve: $$180 \div 6 = 30 \Rightarrow 30 \times 10 = 300$$

Answer: 300 miles

Example 3: Multi-Step Word Problem

Problem: A store applies a 20% discount to an item priced at \$50 and then adds 8% sales tax. What is the final price?

Step 1: Calculate the discount: $$0.20 \times 50 = 10 \Rightarrow 50 - 10 = 40$$

Step 2: Add 8% tax: $$0.08 \times 40 = 3.20 \Rightarrow 40 + 3.20 = 43.20$$

Answer: \$43.20

Self-Study Tip: Word problems often disguise familiar math behind real-world context. Focus on keywords and quantities, draw diagrams if helpful, and always check your answer for reasonableness.

Multimedia Resources

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

https://www.media4math.com/LessonPlans/SupportResourcesSATMathLesson10

Quiz

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

1. A store sells pencils for \$0.50 each. If a customer has \$8.00 and buys as many pencils as possible, how many pencils can they buy?
   a) 14
   b) 15
   c) 16
   d) 18
2. The price of a jacket is increased by 20% and becomes \$96. What was the original price?
   a) \$75
   b) \$78
   c) \$80
   d) \$85
3. John ran 3 miles in 24 minutes. At the same pace, how long will it take him to run 10 miles?
   a) 70 minutes
   b) 75 minutes
   c) 80 minutes
   d) 84 minutes
4. A class has 12 boys and 18 girls. What percent of the class are boys?
   a) 35%
   b) 40%
   c) 45%
   d) 50%
5. During a sale, a laptop originally priced at \$600 is discounted by 15% and then has a 7% tax added. What is the final price?
   a) \$531.60
   b) \$546.00
   c) \$552.90
   d) \$567.00
6. The perimeter of a rectangle is 40 meters. The length is 12 meters. What is the width?
   a) 8
   b) 10
   c) 12
   d) 14
7. A school has a student-to-teacher ratio of 25:1. If there are 600 students, how many teachers are there?
   a) 20
   b) 22
   c) 24
   d) 25
8. Maria spends 40% of her monthly income on rent. If her rent is \$1,000, what is her monthly income?
   a) \$2,200
   b) \$2,400
   c) \$2,500
   d) \$2,800
9. A garden has a length of 15 feet and width of 8 feet. What is the area?
   a) 100 ft²
   b) 110 ft²
   c) 120 ft²
   d) 130 ft²
10. One printer can print a document in 8 minutes, and a second printer can do the same job in 12 minutes. How long will it take both printers working together to complete the job?
    a) 4.8 minutes
    b) 5 minutes
    c) 6 minutes
    d) 6.2 minutes

Answer Key

1. Answer: c) 16
   \$8.00 ÷ \$0.50 = 16 pencils
2. Answer: c) \$80
   Let x be the original price. $1.20x = 96 \Rightarrow x = \frac{96}{1.20} = 80$
3. Answer: c) 80 minutes
   Use a proportion: $\frac{3}{24} = \frac{10}{x} \Rightarrow 3x = 240 \Rightarrow x = 80$
4. Answer: b) 40%
   Total students = 12 + 18 = 30 $\frac{12}{30} \times 100 = 40\%$
5. Answer: c) \$552.90
   15% discount: $0.15 \times 600 = 90 \Rightarrow 600 - 90 = 510$ Add 7% tax: $0.07 \times 510 = 35.70 \Rightarrow 510 + 35.70 = 552.90$
6. Answer: a) 8
   Perimeter = $2L + 2W = 40$ $2(12) + 2W = 40 \Rightarrow 24 + 2W = 40 \Rightarrow 2W = 16 \Rightarrow W = 8$
7. Answer: d) 25
   $\frac{600}{25} = 24$ Correction: **Answer is a) 24** (previous listing said 25, now corrected)
8. Answer: b) \$2,500
   Let x be monthly income: $0.40x = 1,000 \Rightarrow x = \frac{1,000}{0.40} = 2,500$
9. Answer: c) 120 ft²
   Area = $15 \times 8 = 120 \text{ ft}^2$
10. Answer: a) 4.8 minutes
    Work rates: $\frac{1}{8} + \frac{1}{12} = \frac{3}{24} + \frac{2}{24} = \frac{5}{24}$ Time = $\frac{1}{\frac{5}{24}} = \frac{24}{5} = 4.8 \text{ minutes}$