Word Problems
Word problems require translating verbal descriptions into mathematical equations. The key skill is identifying the unknown, writing the equation correctly, and solving systematically.
Fundamental Principles
Age Problems
Involve past, present, and future ages. Let the present age = x and write equations based on given conditions.
Work Problems
If A completes a job in n days, A's rate = 1/n per day. Combined rate = sum of individual rates.
Mixture Problems
Total amount × concentration = amount of substance. Used when mixing solutions of different concentrations.
Distance-Speed-Time
Distance = Speed × Time. If speed changes, track each segment separately.
Essential Formulation Tips
- Always define variables clearly at the start.
- Translate keywords: 'sum' = +, 'difference' = -, 'times' = ×, 'of' = ×, 'is/equals' = =.
- For work problems, use rates (work per unit time) not total work.
- Check your answer by substituting back into the original conditions.
Shortcut Execution Techniques
- For age problems: set up a table (past, present, future) to organize information.
- For mixture: total = parts × concentration for each component.
- Speed upstream = speed of boat - speed of current.
- For consecutive integers: use n, n+1, n+2 (or n-1, n, n+1).
Contextual Inquiries (FAQs)
Q: How do I start a word problem?
A: 1. Read carefully. 2. Identify the unknown. 3. Assign a variable. 4. Write the equation from given conditions. 5. Solve and verify.
Q: What does 'inversely proportional' mean?
A: If y is inversely proportional to x, then y × x = constant, or y = k/x.
Q: How are mixture problems solved?
A: Use the equation: (amount × concentration)₁ + (amount × concentration)₂ = (total amount × final concentration).
Example Breakdown: Age Problem
Classic age problem pattern.Let son's age = x. Father's age = 4x.
In 6 years: 4x + 6 = 3(x + 6)
4x + 6 = 3x + 18
x = 12. Son = 12, Father = 48.
Example Breakdown: Work Problem
Very common in aptitude exams.1/A + 1/B = 1/12
1/20 + 1/B = 1/12
1/B = 1/12 - 1/20 = 5/60 - 3/60 = 2/60 = 1/30
B alone takes 30 days.
Example Breakdown: Mixture Problem
Mixture setup is the same for all concentration problems.Let x = liters of 30% acid added.
0.30x + 0.50(60) = 0.40(x + 60)
0.30x + 30 = 0.40x + 24
6 = 0.10x → x = 60
Word Problems Practice Set 1
Basic word problems: numbers, ages, and simple ratios.
Q1. Twice a number increased by 7 is 23. Find the number.
Q2. The sum of three consecutive even numbers is 78. Find the largest.
Q3. A boy is 4 years older than his sister. If their combined age is 20, how old is the sister?
Q4. A number is 5 less than twice another. Their sum is 31. Find the smaller number.
Q5. Three friends share Rs. 240 in the ratio 3:4:5. What is the largest share?
Q6. The product of two numbers is 48. One is 3 more than the other. Find the larger number.
Q7. If the numerator of a fraction is increased by 3 and denominator by 2, it becomes 7/4. If the original fraction is x/5, find x.
Q8. In 12 years, a man's age will be 3 times his daughter's current age. His daughter is now 7. Find the man's current age.
Q9. The average of 5 numbers is 20. If one number is removed, the average becomes 18. What is the removed number?
Q10. A machine produces 300 items per hour. After an upgrade, it produces 20% more. How many items does it produce in 5 hours post-upgrade?