Solving Word Problems
Word problems are an important part of the ParaPro Math Assessment. They test your ability to translate real-world situations into mathematical equations and solve them correctly.
Understanding Word Problems
Word problems assess how well you can:
- Identify relevant information
- Choose appropriate operations
- Create mathematical models
- Apply mathematical concepts to everyday situations
- Interpret results in context
Step-by-Step Approach to Solving Word Problems
- Read carefully – Understand what the problem is asking
- Identify known and unknown values – What information is given? What are you trying to find?
- Choose a strategy – Decide which mathematical operations or concepts apply
- Set up the problem – Create an equation or model
- Solve – Work through the math systematically
- Check your answer – Verify that your solution makes sense in the context
- Answer the question – Make sure you’re answering what was asked
Common Types of Word Problems
1. Number Problems
Example: Age Problems
Problem: Mary is 5 years older than her brother Tom. If the sum of their ages is 25, how old is Tom?
Solution:
Step 1: Let’s call Tom’s age = x
Step 2: Then Mary’s age = x + 5
Step 3: We know their ages sum to 25, so we write the equation: x + (x + 5) = 25
Step 4: Simplify: 2x + 5 = 25
Step 5: Solve for x: 2x = 20, so x = 10
Step 6: Check: If Tom is 10, then Mary is 10 + 5 = 15. Their sum is 10 + 15 = 25. ✓
Therefore, Tom is 10 years old.
2. Percentage Problems
Example: Discount Calculation
Problem: A classroom set of books originally priced at $480 is on sale for 25% off. How much will the school pay for the discounted set?
Solution:
Step 1: Calculate the discount amount: $480 × 0.25 = $120
Step 2: Subtract the discount from the original price: $480 – $120 = $360
Therefore, the school will pay $360 for the discounted set.
Alternatively, we could have used: $480 × (1 – 0.25) = $480 × 0.75 = $360
3. Rate and Time Problems
Example: Distance Problem
Problem: A school bus travels at 45 miles per hour. How far will it travel in 2.5 hours?
Solution:
Step 1: Use the formula: Distance = Rate × Time
Step 2: Substitute the values: Distance = 45 mph × 2.5 h
Step 3: Calculate: Distance = 112.5 miles
Therefore, the bus will travel 112.5 miles in 2.5 hours.
4. Mixture Problems
Example: Solution Mixture
Problem: A science teacher has two solutions: one is 20% salt and another is 50% salt. How many liters of each solution should be mixed to create 10 liters of a 30% salt solution?
Solution:
Step 1: Let’s call the amount of 20% solution = x liters
Step 2: Then the amount of 50% solution = (10 – x) liters (since total is 10 liters)
Step 3: Set up the equation for the amount of salt: 0.20x + 0.50(10 – x) = 0.30(10)
Step 4: Simplify: 0.20x + 5 – 0.50x = 3
Step 5: Further simplify: -0.30x + 5 = 3
Step 6: Solve for x: -0.30x = -2, so x = 6.67 liters
Step 7: Calculate amount of 50% solution: 10 – 6.67 = 3.33 liters
Therefore, the teacher should mix 6.67 liters of the 20% solution with 3.33 liters of the 50% solution.
Key Phrases in Word Problems
| Operation | Common Phrases |
|---|---|
| Addition | Sum, total, increased by, more than, combined, together |
| Subtraction | Difference, less than, decreased by, fewer, remaining, left over |
| Multiplication | Product, times, multiplied by, of, twice |
| Division | Quotient, divided by, per, out of, ratio |
| Equals | Is, was, amounts to, will be, gives, yields |
Practice Word Problems
Problem 1: A teacher buys 42 notebooks for her class. If each student receives 3 notebooks, how many students are in the class?
Problem 2: A school fundraiser sold 250 tickets. Student tickets cost $5 and adult tickets cost $8. If the total revenue was $1,750, how many adult tickets were sold?
Problem 3: A rectangular playground is 45 feet long and 30 feet wide. What is the cost to install fencing around the perimeter if fencing costs $12 per foot?
Problem 4: The ratio of boys to girls in a class is 3:4. If there are 28 students in total, how many boys are in the class?
Key Points to Remember
- Read the entire problem carefully before trying to solve
- Identify exactly what the problem is asking you to find
- Organize the given information and identify the unknowns
- Select appropriate operations, formulas, or equations
- Solve step by step, keeping track of your work
- Verify your answer makes sense in the context of the problem
- Always include proper units in your answer