Solving Word Problems
The hardest part of word problems isn't the math — it's figuring out what math to do. Here's the systematic approach.
The 5-Step Method
Follow these steps for ANY word problem:
- Step 1: READ — Read the entire problem carefully. Twice.
- Step 2: IDENTIFY — What is the question asking? What are you solving for?
- Step 3: EXTRACT — Write down the given numbers and their meanings.
- Step 4: PLAN — Decide which operation(s) to use.
- Step 5: SOLVE & CHECK — Do the math, then ask: "Does this answer make sense?"
Keyword Clues
Addition Keywords
"total," "sum," "combined," "altogether," "in all," "both," "increased by," "more than," "plus"
Subtraction Keywords
"difference," "remaining," "left over," "less than," "fewer," "decreased by," "how many more," "change"
Multiplication Keywords
"times," "product," "each," "per," "every," "doubled," "tripled," "of" (with fractions/percents)
Division Keywords
"per," "each," "equally," "split," "shared," "average," "ratio," "quotient," "how many groups"
Worked Examples
Example 1: Single-Step
A bakery sold 145 muffins on Monday and 203 on Tuesday. How many total?
Question: total muffins. Operation: addition (keyword "total").
145 + 203 = 348 muffins
Example 2: Multi-Step
You earn $15/hour and work 8 hours. After spending $45 on lunch and supplies, how much do you have?
Step 1: Earnings = $15 × 8 = $120
Step 2: Remaining = $120 - $45 = $75
Example 3: Rate Problem
A printer prints 30 pages per minute. How long to print 450 pages?
Time = Total ÷ Rate = 450 ÷ 30 = 15 minutes
Example 4: Percentage Problem
A laptop costs $800. It's on sale for 25% off. What's the sale price?
Discount = $800 × 0.25 = $200
Sale price = $800 - $200 = $600
Example 5: Working Backward
After a 20% raise, Maria earns $54,000. What was her old salary?
New salary = Old × 1.20. So Old = $54,000 ÷ 1.20 = $45,000
Common Problem Types
Distance-Rate-Time
Distance = Rate × Time. Rearrange as needed:
Rate = Distance ÷ Time. Time = Distance ÷ Rate.
Cost Problems
Total Cost = Unit Price × Quantity
Unit Price = Total Cost ÷ Quantity
Age Problems
Set up relationships between ages now and in the future/past.
Mixture Problems
Total value = (amount₁ × price₁) + (amount₂ × price₂)
The Sanity Check
ALWAYS check your answer:
- Is the answer a reasonable size? (A human can't be 500 years old)
- Are the units correct? (You can't have 5.7 people)
- Does the answer match the question? (Don't give speed when asked for time)
- Can you estimate to verify? (If 32 × 28 should be ~900, and you got 896, you're probably right)
Drawing Pictures
When stuck, draw a picture or diagram. For geometry problems, distance problems, or any spatial problem, a quick sketch can reveal the solution path immediately.
🎯 Try It Yourself
Test your understanding with these practice problems.
1. A train travels at 60 mph. How far does it go in 2.5 hours?
💡 Hint: Distance = speed × time
2. If 5 shirts cost $124.75, how much does one shirt cost?
💡 Hint: Divide total by number of shirts
3. You have $200. After spending 35% on groceries, how much is left?
💡 Hint: $200 × 0.35 = $70 spent, $200 - $70 = ?
4. A pool fills at 50 gallons/hour. It holds 2,000 gallons. How many hours to fill?
💡 Hint: Total ÷ rate = time
5. Ages: Sam is twice as old as Alex. Alex is 14. In 6 years, how old will Sam be?
💡 Hint: Sam is 28 now. In 6 years...
⚠️ Common Mistakes
- ✗Jumping to calculations before understanding what the problem is actually asking for.
- ✗Picking the wrong operation — 'total' usually means multiply or add, 'each/per' usually means divide.
- ✗Ignoring units. If speed is in mph and time is in minutes, convert first!
- ✗Not checking if the answer makes sense. If a problem asks someone's age and you get 347, something went wrong.
- ✗Missing hidden information. 'A dozen' means 12, 'a fortnight' means 14 days, 'twice' means ×2.
🌍 Real Life Example
Road Trip Planning
You're driving from Denver to Las Vegas (750 miles). Your car gets 28 MPG and gas costs $3.50/gallon. How much will gas cost? Gallons needed: 750 ÷ 28 ≈ 26.8 gallons. Cost: 26.8 × $3.50 ≈ $93.75. Driving at an average 65 mph, travel time: 750 ÷ 65 ≈ 11.5 hours. Every road trip is a multi-step word problem!
💡 Key Takeaway
Solving word problems is a 5-step process: (1) Read carefully, (2) Identify what you're solving for, (3) Extract the numbers and relationships, (4) Choose the right operation(s), (5) Solve and check if the answer makes sense. The math is usually the easiest part.