Quick Concept Summary
Chain Rule (Unitary Method) is used when the value of one quantity changes depending on multiple other quantities. You find the value for 1 unit first, then scale to the required quantity.
Two types of proportion:
| Type | Meaning | Example |
|---|---|---|
| Direct | Both increase or both decrease together | More men → more work |
| Inverse | One increases, other decreases | More men → fewer days |
The key rule for setting up a chain:
- Write the unknown quantity on the right
- For each pair of related quantities, write the ratio such that it would make the unknown larger if it should be larger, and smaller if it should be smaller
Key Formulas
Direct Proportion
$$\frac{x_1}{y_1} = \frac{x_2}{y_2}$$
If x₁ = 5 workers earn Rs 200, then x₂ = 8 workers earn: y₂ = 200 × 8/5 = Rs 320
Inverse Proportion
x₁ × y₁ = x₂ × y₂
If 5 workers finish in 20 days, then 10 workers finish in: 5 × 20 / 10 = 10 days
General Chain Rule Formula
For quantities A, B, C where we know values for A₁, B₁, C₁ and need C₂:
$$C_2 = C_1 \times \frac{A_1}{A_2} \text{ (or } \frac{A_2}{A_1}) \times \frac{B_1}{B_2} \text{ (or } \frac{B_2}{B_1})$$
Choose A₁/A₂ or A₂/A₁ based on whether C increases or decreases when A increases.
Worked Examples
Example 1: Direct Proportion
Example 2: Chain Rule with Three Quantities
Example 3: Workers, Hours, and Output
Example 4: Ration / Food Supply Problem
Example 5: Four-Variable Chain
Example 6: Horses and Food
Shortcuts and Tricks
Trick 1: The Proportion Arrow Method
Draw arrows:
- If quantities change in the same direction → multiply (Direct)
- If quantities change in opposite directions → flip ratio (Inverse)
Trick 2: Cancel Before Multiplying
Always cancel common factors across numerators and denominators before multiplying — reduces calculation errors significantly.
Trick 3: Unitary Method Verification
After finding the answer, verify by computing the "1 unit" value:
- Find value per 1 man per 1 day per 1 hour
- Scale up to required quantity
TNPSC Exam Pattern
| Question Type | Frequency | Difficulty |
|---|---|---|
| 3-variable chain (men, days, hours) | Very common | Medium |
| Provisions / ration problems | Common | Medium |
| Earnings with multiple variables | Common | Medium |
| 4-variable chain (pumps, tonnes) | Occasional | Medium–Hard |
Practice Questions
Q1. 8 men can do a job in 15 days. How many days will 12 men take? Answer: 8×15 / 12 = 10 days
Q2. 15 workers earn Rs 4,500 in 6 days. How much will 10 workers earn in 9 days? Answer: 4500 × (10/15) × (9/6) = 4500 × 2/3 × 3/2 = Rs 4,500
Q3. A garrison of 500 men had food for 30 days. After 10 days, 100 men left. How many more days will food last? Answer: Remaining food = 500×20 = 10,000 man-days. Remaining men = 400. Days = 10000/400 = 25 days
Q4. If 3 men or 6 women can reap a field in 20 days, in how many days can 6 men and 3 women reap it? Answer: 1 man = 2 women. 6 men + 3 women = 12 women + 3 women = 15 women. 6 women finish in 20 days → 15 women in 6×20/15 = 8 days
Q5. 10 typists can type 500 pages in 5 days working 6 hours/day. How many typists can type 800 pages in 4 days at 8 hours/day? Answer: Typists = 10 × (800/500) × (5/4) × (6/8) = 10 × 8/5 × 5/4 × 3/4 = 10 × 120/80 = 15 typists
Q6. 16 men working 6 hours/day can build a wall in 10 days. How many days will 24 men at 8 hours/day take? Answer: Days = 10 × (16/24) × (6/8) = 10 × 2/3 × 3/4 = 5 days
Q7. The cost of 25 kg sugar for 30 days for 4 persons is Rs 3,000. What is the cost for 50 kg for 20 days for 6 persons? Answer: Cost = 3000 × (50/25) × (20/30) × (6/4) = 3000 × 2 × 2/3 × 3/2 = Rs 6,000
Q8. 40 cows can graze a field in 16 days. How many cows will graze the same field in 10 days? Answer: 40 × 16 / 10 = 64 cows
Q9. If 18 binders bind 900 books in 10 days, how many binders will bind 660 books in 12 days? Answer: Binders = 18 × (660/900) × (10/12) = 18 × 11/15 × 5/6 = 18 × 55/90 = 11 binders
Q10. A contractor planned to finish a work in 30 days with 40 men. After 10 days, he found only 3/4 of the work was done. How many extra men should he employ? Answer: Work done in 10 days = 3/4 by 40 men. Work remaining = 1/4. Time remaining = 30-10 = 20 days. Men needed = (40 × 10 × 1/4) / (3/4 × 20) ... Let's redo: Total work = 40×30=1200 units. In 10 days by 40 men: 400 units done, but 3/4 done means 900 done — that means more work than expected. Re-read: 3/4 work was NOT done (only 1/4 done). So 3/4 remains. Men = 40×10×(3/4) / (1/4×20) → using M₁D₁/W₁ = M₂D₂/W₂: 40×10/(1/4) = M₂×20/(3/4) → M₂ = 40×10×3/(4×20×1/4) = 1200×3/20 = 60. Extra men = 60-40 = 20