Word Problems
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Word problems are the bridge between abstract mathematics and real-world applications. This lesson focuses on developing systematic strategies for translating verbal descriptions into mathematical models and solving them effectively.
Word Problem Strategy: GUESSS
Use this systematic approach for complex word problems:
- Given - What information is provided?
- Unknown - What are you solving for?
- Equation - What mathematical relationship applies?
- Solve - Perform the calculations
- State - Express the answer in context
- Sanity check - Does the answer make sense?
Common Word Problem Categories
- Rate problems - Work, distance, flow rates
- Mixture problems - Concentrations, blending
- Financial problems - Interest, profit, cost
- Geometric problems - Area, volume, dimensions
- Growth/decay problems - Population, radioactivity
Key Phrases and Their Mathematical Meanings
| Phrase | Mathematical Operation |
|---|---|
| "increased by," "more than," "sum of" | Addition (+) |
| "decreased by," "less than," "difference" | Subtraction (-) |
| "of," "times," "product of" | Multiplication (*) |
| "per," "ratio," "quotient" | Division (/) |
| "is," "equals," "results in" | Equals (=) |
| "at least" | Greater than or equal |
| "at most" | Less than or equal |
Examples
Example 1: Work Rate Problem
Problem: Pipe A can fill a tank in 6 hours. Pipe B can fill it in 4 hours. How long will it take both pipes working together?
Given: Pipe A rate = 1/6 tank per hour; Pipe B rate = 1/4 tank per hour
Unknown: Time t when combined rate fills 1 tank
Equation: (1/6 + 1/4) * t = 1
Solve: (2/12 + 3/12) * t = 1; (5/12) * t = 1; t = 12/5 = 2.4 hours
State: Working together, the pipes fill the tank in 2 hours 24 minutes.
Sanity check: Less than either individual time, which makes sense.
Example 2: Age Problem
Problem: Sarah is 4 times as old as her son. In 20 years, she will be twice as old as her son. How old is each now?
Given: Sarah = 4 * Son (now); Sarah + 20 = 2(Son + 20)
Unknown: Son's current age (s), Sarah's current age (4s)
Equation: 4s + 20 = 2(s + 20)
Solve: 4s + 20 = 2s + 40; 2s = 20; s = 10
State: Son is 10, Sarah is 40.
Check: In 20 years: Son = 30, Sarah = 60. 60 = 2(30). Correct!
Practice Problems
Apply the GUESSS strategy to each problem. Show all steps.
Problem 1: Distance-Rate-Time
A cyclist travels from Town A to Town B at 20 mph and returns at 12 mph. If the total trip took 8 hours, how far apart are the towns?
Problem 2: Mixture
A chemist needs 100 mL of a 40% alcohol solution. She has 25% and 50% solutions available. How much of each should she mix?
Problem 3: Investment
An investor has $20,000 to invest. Part is invested at 5% and the rest at 8%. If the total annual interest is $1,300, how much was invested at each rate?
Problem 4: Work Rate
Machine A can complete a job in 10 hours. Machine B can complete it in 15 hours. After Machine A works alone for 4 hours, Machine B joins. How much longer until the job is complete?
Problem 5: Geometry
The length of a rectangle is 3 more than twice its width. If the perimeter is 48 cm, find the dimensions and area.
Problem 6: Motion (Same Direction)
A freight train leaves a station at 8 AM traveling at 40 mph. A passenger train leaves the same station at 10 AM traveling at 60 mph in the same direction. At what time will the passenger train catch up?
Problem 7: Consecutive Integers
The sum of three consecutive odd integers is 111. Find the integers.
Problem 8: Percent Change
A store marks up items by 40% and then offers a 25% discount on the marked price. What is the overall percent change from the original cost?
Problem 9: Exponential Decay
A radioactive substance has a half-life of 5 days. If you start with 200 grams, how much remains after 12 days?
Problem 10: Systems Application
Movie tickets cost $12 for adults and $8 for children. A group of 25 people spent $260 on tickets. How many adults and how many children were in the group?
Problem 11: Optimization
A farmer has 200 meters of fencing to enclose a rectangular area against a barn (so only 3 sides need fencing). What dimensions maximize the enclosed area?
Problem 12: Rate of Change
Water is draining from a tank at a rate modeled by V(t) = 1000 - 50t + t^2, where V is in gallons and t is in minutes. When will the tank be empty, and what is the average rate of drainage?
Check Your Understanding
Self-Assessment Questions
- Can you identify the type of word problem (rate, mixture, financial, etc.)?
- Are you able to define variables clearly before writing equations?
- Do you check your answers against the original problem conditions?
- Can you recognize when a problem requires a system of equations?
Answer Key
- Distance = 30 miles
- 40 mL of 25% solution, 60 mL of 50% solution
- $10,000 at 5%, $10,000 at 8%
- 3.6 more hours (job done at 7.6 hours total)
- Width = 7 cm, Length = 17 cm, Area = 119 sq cm
- 2 PM (4 hours after passenger train departs)
- 35, 37, 39
- 5% increase overall (1.40 * 0.75 = 1.05)
- Approximately 38.4 grams
- 15 adults, 10 children
- Width = 50 m, Length = 100 m, Area = 5000 sq m
- t = 50 minutes (using quadratic formula); Average rate = 20 gallons/minute
Next Steps
- Create a personal reference sheet of problem types and strategies
- Practice translating without solving to build equation-writing skills
- Review Common Mistakes to avoid typical errors
- Time yourself on problems to build speed for standardized tests