Grade: Grade 11 Subject: Mathematics Unit: Algebra II Completion Lesson: 4 of 6 SAT: AdvancedMath ACT: Math

Word Problems

Translate real-world scenarios into algebraic expressions and equations, then solve using the techniques you have learned.

Learn

Word problems require you to extract mathematical relationships from written descriptions. This skill is essential for SAT and ACT success, where many problems are presented in context.

Word Problem Strategy (UPSC)

  1. Understand: Read the problem carefully. Identify what you are asked to find.
  2. Plan: Assign variables and write equations that model the situation.
  3. Solve: Use algebraic techniques to find the solution.
  4. Check: Verify your answer makes sense in the context of the problem.

Common Word Problem Types in Algebra II

  • Optimization: Finding maximum or minimum values (often using quadratics)
  • Rate and Work: Problems involving speed, time, and combined rates
  • Growth and Decay: Exponential models for populations, investments, radioactive decay
  • Geometry Applications: Area, volume, and dimensional relationships
  • Mixture Problems: Combining substances with different concentrations

Key Phrases to Mathematical Translations

  • "is," "equals," "results in" translates to =
  • "more than," "increased by" translates to +
  • "less than," "decreased by" translates to -
  • "product of," "times" translates to multiplication
  • "quotient of," "divided by" translates to division
  • "at most" translates to less than or equal
  • "at least" translates to greater than or equal

Examples

Study these worked examples to see how to translate and solve word problems.

Example 1: Projectile Motion

Problem: A ball is thrown upward from a height of 6 feet with an initial velocity of 64 feet per second. The height h of the ball after t seconds is given by h(t) = -16t^2 + 64t + 6. What is the maximum height reached by the ball?

Solution:

  1. The maximum occurs at the vertex of the parabola
  2. Vertex x-coordinate: t = -b/(2a) = -64/(2(-16)) = 2 seconds
  3. Maximum height: h(2) = -16(4) + 64(2) + 6 = -64 + 128 + 6 = 70 feet

Example 2: Work Rate Problem

Problem: Alex can paint a room in 4 hours. Jordan can paint the same room in 6 hours. How long will it take them to paint the room together?

Solution:

  1. Alex's rate: 1/4 room per hour
  2. Jordan's rate: 1/6 room per hour
  3. Combined rate: 1/4 + 1/6 = 3/12 + 2/12 = 5/12 room per hour
  4. Time to complete: 1 / (5/12) = 12/5 = 2.4 hours (2 hours 24 minutes)

Practice

Apply the UPSC strategy to solve these word problems. Show all your work.

Problem 1: A rectangular garden has a perimeter of 56 feet. If the length is 4 feet more than twice the width, find the dimensions of the garden.

Problem 2: The profit P from selling x units of a product is given by P(x) = -2x^2 + 120x - 800. How many units should be sold to maximize profit, and what is the maximum profit?

Problem 3: A car travels 120 miles at a constant speed. If the car had traveled 10 mph faster, the trip would have taken 30 minutes less. What was the car's original speed?

Problem 4: An investment of $5,000 earns compound interest at an annual rate of 6%. Write a function for the value after t years, and find how long it takes to double (use the formula A = P(1 + r)^t).

Problem 5: Two pipes can fill a tank together in 12 hours. The larger pipe alone can fill the tank in 20 hours. How long would it take the smaller pipe alone to fill the tank?

Problem 6: A farmer wants to enclose a rectangular field along a river, using 800 feet of fencing for the three sides not along the river. What dimensions will maximize the enclosed area?

Problem 7: The sum of two numbers is 20. The sum of their squares is 232. Find both numbers.

Problem 8: A radioactive substance decays according to the formula A(t) = A_0 * (0.5)^(t/h), where h is the half-life. If a sample decreases from 100 grams to 25 grams in 10 years, what is the half-life?

Problem 9: A boat travels 24 miles upstream and 24 miles downstream. The trip upstream takes 3 hours and the trip downstream takes 2 hours. Find the speed of the boat in still water and the speed of the current.

Problem 10: A chemist has two solutions: one is 30% acid and the other is 60% acid. How many liters of each should be mixed to obtain 30 liters of a 50% acid solution?

Check Your Understanding

Reflect on the strategies and techniques used in this lesson.

  1. What is the first step you should always take when solving a word problem?
  2. How do you find the maximum or minimum value of a quadratic function in a word problem context?
  3. Explain how to set up a rate-of-work equation when two people or machines work together.
  4. Why is it important to check your answer in the context of the original problem?

Next Steps

  • Review any problem types that were challenging
  • Practice translating phrases into algebraic expressions
  • Move on to Common Mistakes to learn what errors to avoid
  • Consider creating your own word problems to deepen understanding