Word Problems
Master the art of translating real-world scenarios into mathematical equations and solving them step by step.
What You'll Learn
- ✓ Develop a systematic approach to solving word problems
- ✓ Identify keywords that indicate mathematical operations
- ✓ Translate word problems into mathematical equations
- ✓ Solve various types of real-world problems
- ✓ Help students approach word problems with confidence
The Problem-Solving Process
The CUBES Method
CUBES is a systematic approach to solving word problems:
C - Circle the numbers
Find and mark all numerical values in the problem
U - Underline the question
Identify exactly what you need to find
B - Box keywords
Mark words that indicate operations (sum, difference, product, etc.)
E - Eliminate extra information
Cross out unnecessary details
S - Solve and check
Work through the problem and verify your answer makes sense
Alternative Step-by-Step Approach
- 1 Read carefully - Read the entire problem at least twice
- 2 Identify what's given - List all known information
- 3 Determine what's needed - What are you solving for?
- 4 Choose a strategy - Select the appropriate operation(s)
- 5 Set up the equation - Translate words to math
- 6 Solve - Work through the calculations
- 7 Check - Does the answer make sense?
Keywords and Operations
Operation Keywords Guide
Addition (+)
- • sum, total, combined
- • plus, added to, increased by
- • more than, greater than
- • altogether, in all
- • and, both
Subtraction (−)
- • difference, minus, subtract
- • less than, fewer than
- • decreased by, reduced by
- • take away, left, remain
- • how many more/fewer
Multiplication (×)
- • product, times, multiplied by
- • of (often with fractions)
- • twice, double, triple
- • groups of, sets of
- • each, every, per
Division (÷)
- • quotient, divided by
- • split, shared equally
- • per, each, average
- • half, third, quarter
- • ratio, rate
⚠️ Watch for tricky phrases:
• "5 less than x" means x − 5, not 5 − x
• "3 times as many as" indicates multiplication
• "divided into" can mean division in reverse order
Basic Operation Problems
Addition Example
Problem: Maria has 24 stickers. Her friend gives her 15 more stickers. How many stickers does Maria have in total?
Solution:
- • Starting stickers: 24
- • Additional stickers: 15
- • Total = 24 + 15 = 39 stickers
Subtraction Example
Problem: A store had 85 apples. They sold 37 apples. How many apples are left?
Solution:
- • Starting apples: 85
- • Apples sold: 37
- • Remaining = 85 - 37 = 48 apples
Multiplication Example
Problem: Each classroom needs 25 chairs. If there are 8 classrooms, how many chairs are needed in total?
Solution:
- • Chairs per classroom: 25
- • Number of classrooms: 8
- • Total = 25 × 8 = 200 chairs
Division Example
Problem: 72 students need to be divided equally into 6 groups. How many students will be in each group?
Solution:
- • Total students: 72
- • Number of groups: 6
- • Students per group = 72 ÷ 6 = 12 students
Multi-Step Word Problems
Example 1: Shopping Problem
Jake buys 3 notebooks for $4 each and 2 pens for $2 each. If he pays with a $20 bill, how much change does he receive?
Solution:
- 1. Cost of notebooks: 3 × $4 = $12
- 2. Cost of pens: 2 × $2 = $4
- 3. Total cost: $12 + $4 = $16
- 4. Change: $20 - $16 = $4
Example 2: Time Problem
A movie starts at 2:15 PM and lasts for 2 hours and 25 minutes. There is a 15-minute intermission in the middle. What time does the movie end?
Solution:
- 1. Movie duration: 2 hours 25 minutes
- 2. Intermission: 15 minutes
- 3. Total time: 2h 25m + 15m = 2h 40m
- 4. End time: 2:15 PM + 2h 40m = 4:55 PM
Example 3: Distance Problem
A family drives 150 miles to visit relatives, then continues 85 miles to a vacation spot. On the return trip, they take a shortcut that is 195 miles. How much shorter is the return trip?
Solution:
- 1. Trip there: 150 + 85 = 235 miles
- 2. Return trip: 195 miles
- 3. Difference: 235 - 195 = 40 miles shorter
Common Problem Types
Money Problems
Key Strategies:
- • Convert all amounts to the same unit
- • Track spending and remaining money
- • Check that answers make sense
Example: Sarah buys items costing $3.45, $2.30, and $5.15. She pays with $20.
Total: $10.90 | Change: $9.10
Time Problems
Key Strategies:
- • Convert between hours and minutes
- • Use a timeline for complex scheduling
- • Remember: 60 minutes = 1 hour
Example: John needs 45 min to get ready and 30 min to drive. Arrive by 9:00 AM.
Total: 75 min = 1h 15m | Wake up: 7:45 AM
Distance/Rate Problems
Distance = Rate × Time
Also: Rate = Distance ÷ Time | Time = Distance ÷ Rate
Example: A car travels 240 miles in 4 hours.
Rate = 240 ÷ 4 = 60 miles per hour
Problem-Solving Strategies
Draw a Picture
Visual representations help understand relationships between quantities. Draw diagrams, charts, or simple sketches.
Make a Table
Organize information in rows and columns to see patterns and relationships clearly.
Work Backwards
Start with the final result and reverse the operations to find the starting value.
Guess and Check
Make educated guesses and refine them based on whether they're too high or too low.
Checking Your Answer
Ask yourself:
- ✓ Does my answer make sense in the context?
- ✓ Is the answer reasonable? (Not too big or too small)
- ✓ Did I answer the question that was asked?
- ✓ Can I verify by working backwards?
- ✓ Are my units correct?
Common Mistakes and How to Avoid Them
Mistake 1: Not reading carefully
Missing important details or misunderstanding what's being asked
✓ Solution: Read the problem at least twice, underline key information
Mistake 2: Using the wrong operation
Confusing "less than" with subtraction order or "of" with addition
✓ Solution: Circle operation keywords and double-check their meaning
Mistake 3: Computational errors
Making arithmetic mistakes during calculations
✓ Solution: Show all work, check calculations, estimate first
Mistake 4: Answering the wrong question
Solving for an intermediate step instead of the final answer
✓ Solution: Reread the question before writing your final answer
Practice Problems
Basic Problems
1. A baker made 48 cookies in the morning and 36 cookies in the afternoon. How many cookies did the baker make in total?
Show Answer
48 + 36 = 84 cookies (addition - "total")
2. There are 156 students going on a field trip. Each bus can hold 52 students. How many buses are needed?
Show Answer
156 ÷ 52 = 3 buses (division - equal groups)
3. Lisa saves $15 each week. How much money will she have saved after 8 weeks?
Show Answer
$15 × 8 = $120 (multiplication - repeated addition)
Multi-Step Problems
4. Tom buys 4 packs of pencils with 12 pencils in each pack. He gives 15 pencils to his sister. How many pencils does Tom have left?
Show Answer
Step 1: 4 × 12 = 48 pencils total
Step 2: 48 - 15 = 33 pencils left
5. A store has 250 apples. They sell 85 apples on Monday and 92 apples on Tuesday. How many apples remain?
Show Answer
Step 1: 85 + 92 = 177 apples sold
Step 2: 250 - 177 = 73 apples remain
6. Movie tickets cost $8 for children and $12 for adults. What is the total cost for 3 children and 2 adults?
Show Answer
Children: 3 × $8 = $24
Adults: 2 × $12 = $24
Total: $24 + $24 = $48
Key Takeaways
- ✓ Read Carefully: Understanding the problem is half the solution
- ✓ Identify Keywords: Words signal which operations to use
- ✓ Plan Before Solving: Organize information and choose a strategy
- ✓ Check Your Work: Verify that your answer makes sense
- ✓ Practice Regularly: Word problems get easier with experience
Related Lessons
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Excellent work mastering word problems! These problem-solving skills will help you in all areas of mathematics and in real-life situations.
