Ratio & Proportion – Rajasthan Patwari / Lekhpal (Quantitative Aptitude)
Why it matters?
Patwari/Lekhpal exams test your ability to handle quick calculations that involve ratios, proportions, percentages, and their variations. Mastery of this topic gives you an edge in Section‑II: Numerical Ability and Section‑III: Data Interpretation.
1. Simple Explanation
| Concept | Definition | Quick Visual |
|---|---|---|
| Ratio | A comparison of two numbers by division. | 3 : 4 → 3/4 |
| Proportion | An equation that states two ratios are equal. | a : b = c : d |
| Direct Proportion | As one value increases, the other increases proportionally. | y = kx |
| Inverse Proportion | As one value increases, the other decreases proportionally. | y = k/x |
Key Idea: If
a : b = c : d, thenad = bc(cross‑multiplication).
2. Key Points / Cheatsheet
| Topic | Formula / Rule | Practical Use |
|---|---|---|
| Ratio | a : b = a/b |
Quick comparison |
| Proportion | a : b = c : d → ad = bc |
Solve for missing term |
| Direct Proportion | y/x = k → y = kx |
Scaling: 3:5 = 9:15 |
| Inverse Proportion | xy = k → y = k/x |
Speed‑distance: speed : time = distance : 1 |
| Percent Increase / Decrease | new = old × (1 ± p/100) |
Price changes |
| Finding Missing Ratio Term | If a : b = c : ? → ? = (b × c)/a |
Find missing part |
| When Two Ratios are Equal | a:b = c:d → a/b = c/d → cross‑multiply |
Check equality quickly |
| Ratios in Different Forms | a:b = (a×k) : (b×k) (k any non‑zero) |
Scaling up/down |
3. Worked Example
Problem (Patwari style):
The land area of Village A is 120 ha and Village B is 180 ha. The tax rates per hectare are ₹200 for A and ₹300 for B. What is the total tax revenue from both villages?
Step‑by‑Step Solution
| Step | Explanation | Calculation |
|---|---|---|
| 1. Find tax for Village A | Area × Rate |
120 ha × ₹200/ha = ₹24,000 |
| 2. Find tax for Village B | Area × Rate |
180 ha × ₹300/ha = ₹54,000 |
| 3. Add both taxes | Total revenue | ₹24,000 + ₹54,000 = ₹78,000 |
Answer: ₹78,000
Why Ratio?
- Ratio of areas:
120 : 180 = 2 : 3. - Ratio of rates:
200 : 300 = 2 : 3. - Since both ratios are equal, the total tax is in the same ratio:
₹24,000 : ₹54,000 = 2 : 3. - That confirms the calculation without adding:
Total = ₹24,000 + 3 × ₹24,000 = ₹78,000.
4. Quick Revision Summary
| Topic | Formula | Quick Tip |
|---|---|---|
| Ratio | a : b = a/b |
Remember : means “to” → a to b |
| Proportion | a : b = c : d ⇔ ad = bc |
Cross‑multiply to solve |
| Direct Proportion | y = kx (k = constant) |
y/x is constant |
| Inverse Proportion | xy = k (k = constant) |
x × y is constant |
| Percent Change | new = old × (1 ± p/100) |
Use ± for increase/decrease |
| Scaling Ratios | a:b = ka:kb |
Multiply by any k to keep ratio same |
| Common Test Pattern | “Find the missing term in a ratio”, “Find equivalent ratio”, “Check if two ratios are equal” | Always cross‑multiply first |
5. Quick Practice Questions (Try before the exam)
- Direct Proportion – If 5 workers finish a job in 12 days, how many days will 8 workers finish the same job?
- Inverse Proportion – A machine prints 150 pages in 5 minutes. How many minutes will it take to print 450 pages?
- Missing Ratio Term –
x : 9 = 12 : 15– Findx. - Percentage – A plot of 200 m² is sold at a 20% profit. What is the selling price if the cost price is ₹4,000?
- Ratio Equality – Check if
7 : 9equals14 : 18.
Answer Key (short)
- 9 days (since
days ∝ 1/workers)- 15 minutes (
time ∝ pages)x = 8- ₹4,800
- Yes, both simplify to
7 : 9
6. Final Takeaway
- Always reduce ratios to simplest form before comparison.
- Cross‑multiplication is the fastest way to verify or solve proportions.
- Keep constants handy: For direct proportion,
k = y/x; for inverse,k = xy. - Practice writing ratios in different forms (fractions, percentages, decimals) – the exam may ask for any.
Good luck with your preparation!