Grade: Grade 12 Subject: Mathematics Unit: Quantitative Reasoning Lesson: 4 of 6 SAT: ProblemSolving+DataAnalysis ACT: Math

Word Problems

Learn

Word problems present quantitative reasoning challenges in real-world contexts. This lesson focuses on translating verbal descriptions into mathematical operations and interpreting results meaningfully.

The UPSC Method for Word Problems

  • U - Understand: Read the problem twice. What are you trying to find? What information is given?
  • P - Plan: Decide what operations or formulas you need. Draw a diagram if helpful.
  • S - Solve: Execute your plan step by step. Show your work clearly.
  • C - Check: Does your answer make sense? Did you answer what was asked?

Common Word Problem Categories

  • Rate problems: Distance/time, work rates, cost per unit
  • Mixture problems: Combining solutions, averaging quantities
  • Growth/decay: Population, investment, depreciation
  • Comparison problems: Ratios, percentages, relative change
  • Optimization: Finding maximum or minimum values

Key Phrases and Their Mathematical Meanings

  • "Total" or "sum of" = addition
  • "Difference" or "how much more" = subtraction
  • "Product" or "times" = multiplication
  • "Per" or "for each" = division or rate
  • "Percent of" = multiply by decimal
  • "Is" or "equals" = equals sign

Examples

Work through these word problems using the UPSC method.

Example 1: Rate Problem

Problem: A factory produces 240 widgets in 6 hours. At this rate, how many hours will it take to produce 800 widgets?

Understand: Find time needed for 800 widgets. Rate is 240 widgets per 6 hours.

Plan: Find rate per hour, then divide 800 by that rate.

Solve: Rate = 240/6 = 40 widgets/hour. Time = 800/40 = 20 hours.

Check: 40 widgets/hour x 20 hours = 800 widgets. Correct!

Answer: 20 hours

Example 2: Mixture Problem

Problem: A store mixes coffee worth $12/lb with coffee worth $8/lb. If 20 lbs of the $12 coffee is used, how many lbs of $8 coffee is needed for a mixture worth $9.50/lb?

Understand: Find pounds of cheaper coffee. Total value must average $9.50/lb.

Plan: Let x = lbs of $8 coffee. Set up value equation.

Solve: 12(20) + 8(x) = 9.50(20 + x)

240 + 8x = 190 + 9.5x

50 = 1.5x

x = 33.33 lbs

Check: Total value = 240 + 266.67 = 506.67. Total weight = 53.33 lbs. Rate = 9.50/lb. Correct!

Answer: Approximately 33.3 pounds of $8 coffee

Example 3: Growth Problem

Problem: A city's population is 125,000 and grows 3.5% annually. What will the population be in 4 years?

Understand: Find future population with compound growth.

Plan: Use formula P = P0(1 + r)^t

Solve: P = 125,000(1.035)^4 = 125,000(1.1475) = 143,438

Check: Each year grows by about 3.5%, so after 4 years roughly 14% growth. 125,000 x 1.14 = 142,500. Close to our answer.

Answer: Approximately 143,438 people

Practice

Apply the UPSC method to solve these word problems.

Problem 1: A train travels 450 miles at an average speed of 60 mph. A second train travels the same distance at 75 mph. How much time does the faster train save?

Problem 2: Sarah invests $5,000 at 5% annual compound interest. Tom invests $5,500 at 3% simple interest. Who has more money after 6 years, and by how much?

Problem 3: A store marks up wholesale prices by 40%. During a sale, items are discounted 25% off the marked price. What is the store's actual profit percentage over wholesale?

Problem 4: Three friends split a dinner bill. If Alex pays 40% of the total, Bailey pays $35, and Casey pays the remaining $25, what was the total bill?

Problem 5: A car depreciates 15% per year. If it was worth $32,000 new, what is its value after 3 years?

Problem 6: Worker A can complete a job in 8 hours. Worker B can complete the same job in 12 hours. How long will it take them working together?

Problem 7: A pool can be filled by Pipe A in 4 hours or by Pipe B in 6 hours. If both pipes are open but there is a drain that empties the pool in 12 hours, how long to fill the pool?

Problem 8: The average of five numbers is 72. If two of the numbers are 65 and 80, and the remaining three numbers are equal, what is the value of each of those three numbers?

Problem 9: A rectangular garden has a perimeter of 56 feet. If the length is 4 feet more than twice the width, what are the dimensions?

Problem 10: Store A sells a product for $120 with a 10% discount. Store B sells the same product for $100 but charges 8% tax. Which store offers the better deal, and by how much?

Problem 11: A chemist needs to dilute a 60% acid solution to make 500 mL of 36% solution. How many mL of pure water must be added?

Problem 12: An investor puts $10,000 into two accounts. One earns 4% and the other earns 7%. If the total interest after one year is $580, how much was invested in each account?

Check Your Understanding

Test yourself with these review questions.

Question 1: What does the "U" in the UPSC method stand for, and why is it important?

Question 2: When you see the phrase "per unit" in a word problem, what operation does this typically indicate?

Question 3: Describe the difference between a rate problem and a mixture problem.

Next Steps

  • Practice translating word problems into equations daily
  • Proceed to Common Mistakes to learn what errors to avoid
  • Keep a problem-solving journal to track your approaches