Solving real-world problems using ratio and proportion concepts to find unknown quantities and comparative relationships.
## Core concept
Word problems involving ratio and proportion require you to: - Translate everyday language into mathematical relationships - Identify whether the problem involves simple ratio, compound ratio, direct proportion, or inverse proportion - Set up equations correctly using the given ratios - Solve for unknowns systematically
The key is recognizing what is being compared (quantities) and how they relate (in a fixed ratio or proportional relationship).
### Direct vs Inverse scenarios - Direct proportion (a ∝ b): As one increases, the other increases proportionally. Example: cost increases as quantity increases. - Inverse proportion (a ∝ 1/b): As one increases, the other decreases proportionally. Example: time to complete a job decreases as number of workers increases.
## Formula / rule
Setting up ratio equations:
If quantities A and B are in ratio m : n, then: - A/B = m/n - A = (m/(m+n)) × Total (when finding part from whole) - B = (n/(m+n)) × Total
Direct proportion: - If a : b = c : d (both in same ratio), then a/b = c/d
Inverse proportion: - If a and b are inversely proportional: a × b = constant - If a₁ and b₁ are one pair: a₁ × b₁ = a₂ × b₂
## Worked example
Problem: Three partners A, B, and C invest money in the ratio 3 : 4 : 5. The total profit is ₹7,200. Calculate each partner's share.
Solution: - Ratio parts: 3 + 4 + 5 = 12 - A's share = (3/12) × 7,200 = ₹1,800 - B's share = (4/12) × 7,200 = ₹2,400 - C's share = (5/12) × 7,200 = ₹3,000
Verification: 1,800 + 2,400 + 3,000 = ₹7,200 ✓
## Common exam applications
- Profit/Loss distribution — Partners invest in given ratios; divide final profit accordingly
- Mixtures and alloys — Quantities of ingredients in specified ratios; find amounts
- Work problems — If A and B work in ratio of efficiencies, calculate time or work done
- Speed and distance — Two vehicles travel at different speeds; find distance or time
- Cost allocation — Divide total cost among items in given ratio
- Population or resource sharing — Distribute resources among groups in given proportion
Typical question pattern: - "If A : B = 5 : 3 and their difference is ₹600, find A and B" - "Workers A and B complete a job in 10 and 15 days respectively. How long together?" - "A mixture contains milk and water in ratio 7 : 2. If milk is 14 litres, find water"
## Common mistakes
- Forgetting to find the total ratio sum — Always add all ratio parts before calculating individual shares
- Confusing which quantity is numerator — "A : B = 3 : 4" means A/B = 3/4, not 4/3
- Not identifying proportion type — Mistaking inverse for direct (or vice versa) leads to wrong answer
- Incomplete verification — Always check that individual parts sum to total or that ratios are satisfied
- Setting up wrong equation for inverse problems — Remember a × b = k (constant), not a/b
- Misreading the problem — "Remaining" or "excess" requires subtraction before applying ratio
Exam strategy: Label unknowns clearly (e.g., let A's share = 3x, B's share = 4x), substitute into constraints, solve for x, then find individual values.