Word Problems
Apply proportional relationships to solve real-world problems.
Learn
Proportional relationships appear in many real-world situations. Learning to recognize and solve these problems is an essential skill.
Common Proportional Situations
- Unit rates: Price per item, speed, miles per gallon
- Scaling: Maps, blueprints, recipes
- Percentages: Discounts, tips, taxes
- Similar figures: Shadows, enlargements
Problem-Solving Strategy
- Identify what quantities are proportional
- Set up a proportion: a/b = c/d
- Cross-multiply to solve
- Check that your answer makes sense
Practice
Question 1: A car travels 150 miles on 5 gallons of gas. How many gallons are needed to travel 420 miles?
Answer
Set up: 150/5 = 420/x. Cross multiply: 150x = 2100. Solve: x = 14 gallons.
Question 2: A recipe calls for 2 cups of flour to make 24 cookies. How many cups are needed for 60 cookies?
Answer
2/24 = x/60. Cross multiply: 24x = 120. Solve: x = 5 cups of flour.
Question 3: On a map, 1 inch represents 25 miles. If two cities are 3.5 inches apart on the map, what is the actual distance?
Answer
1/25 = 3.5/x. Cross multiply: x = 87.5 miles.
Question 4: A 6-foot tall person casts a 4-foot shadow. How tall is a tree that casts a 20-foot shadow?
Answer
6/4 = x/20. Cross multiply: 4x = 120. Solve: x = 30 feet tall.
Question 5: If 3 workers can complete a job in 12 hours, how long would it take 4 workers? (Assume the relationship is inversely proportional.)
Answer
This is inverse proportion. 3 x 12 = 4 x t. 36 = 4t. t = 9 hours. More workers means less time.
Question 6: A store has a 20% off sale. If a jacket costs $75, what is the sale price?
Answer
20% of $75 = 0.20 x 75 = $15 discount. Sale price = $75 - $15 = $60.
Question 7: A jogger runs 3 miles in 27 minutes. At this rate, how long will it take to run 5 miles?
Answer
3/27 = 5/x. Cross multiply: 3x = 135. Solve: x = 45 minutes.
Question 8: If 8 pencils cost $2.40, how much do 12 pencils cost?
Answer
8/2.40 = 12/x. Cross multiply: 8x = 28.80. Solve: x = $3.60.
Question 9: A blueprint uses a scale of 1 cm = 2 feet. If a room is 6 cm wide on the blueprint, what is the actual width?
Answer
1/2 = 6/x. Cross multiply: x = 12 feet.
Question 10: Sarah earned $45 for 6 hours of work. How much will she earn for 10 hours at the same rate?
Answer
45/6 = x/10. Cross multiply: 6x = 450. Solve: x = $75.
Check Your Understanding
- How do you identify a proportional relationship in a word problem?
- What is the difference between direct and inverse proportion?
- How do you check if your answer makes sense?
Next Steps
- Practice identifying proportional relationships in everyday situations
- Continue to Lesson 5: Common Mistakes