1. Unitary method
Find the value for one unit, then multiply. Decide first whether the quantities are in direct or inverse proportion.
- Direct: more items → more cost.
- Inverse: more men → fewer days.
2. Chain rule (M D H / W)
M₁ × D₁ × H₁ / W₁ = M₂ × D₂ × H₂ / W₂ (men, days, hours per day, work).
15 men build a wall in 20 days; 10 men take 15 × 20/10 = 30 days.
10 workers make 30 toys in 6 days; 15 workers make 75 toys in 10 × 6 × 75/(15 × 30) = 10 days.
3. Fort and food
Food = men × days. Subtract what has been eaten, then divide by the new number of men.
(ASO 2015) Food for 300 men for 90 days; after 20 days 50 men leave → 300 × 70/250 = 84 days.
4. Wages
Wages are shared in the ratio of work done, which is the ratio of efficiencies (1/a : 1/b), not days.
A in 10 days, B in 15 days earn ₹1,500 together → shares 15 : 10 = 3 : 2 → A gets ₹900.
Solved examples by question type
Each type that comes in the ASO paper, worked step by step. Try it yourself first, then read the solution.
ASO exam level
Example 1. A fort has food for 300 men for 90 days. After 10 days, 50 men leave. How long will the remaining food last for the rest?
(a) 96 days (b) 106 days (c) 80 days (d) 108 days
Answer: (a) 96 days
Solution: Food left = 300 × 80 = 24000 man-days. Men now = 250. Days = 24000/250 = 96.
Example 2. 10 men working 8 hours a day finish a job in 12 days. In how many days will 12 men working 8 hours a day finish it?
(a) 14 2/5 days (b) 14 days (c) 12 days (d) 10 days
Answer: (d) 10 days
Solution: M₁D₁H₁ = M₂D₂H₂ → 10×12×8 = 12×D×8 → D = 10 days.
Example 3. 12 men working 8 hours a day finish a job in 15 days. In how many days will 20 men working 9 hours a day finish it?
(a) 22 2/9 days (b) 9 days (c) 10 days (d) 8 days
Answer: (d) 8 days
Solution: M₁D₁H₁ = M₂D₂H₂ → 12×15×8 = 20×D×9 → D = 8 days.
Example 4. 6 workers make 60 toys in 10 days. How many days will 15 workers take to make 75 toys?
(a) 20 days (b) 8 days (c) 12 1/2 days (d) 5 days
Answer: (d) 5 days
Solution: Work ∝ men × days: 6×10/60 = 15×D/75 → D = 5 days.
Example 5. A can do a work in 10 days and B in 15 days. Together they finish it and get ₹3,000. A's share is:
(a) ₹1,200 (b) ₹1,500 (c) ₹1,800 (d) ₹1,900
Answer: (c) ₹1,800
Solution: Wages are shared in the ratio of work done = efficiency ratio = 1/10 : 1/15 = 15 : 10. A = 3,000 × 15/25 = ₹1,800.
Example 6. A, B and C can do a job in 10, 15 and 30 days. Working together they earn ₹3,600. C's share is:
(a) ₹1,200 (b) ₹600 (c) ₹750 (d) ₹1,800
Answer: (b) ₹600
Solution: Efficiencies 3 : 2 : 1. C = 3,600 × 1/6 = ₹600.
Advanced level
Example 7. If x men working x hours a day can do x units of work in x days, how many units will y men working y hours a day do in y days?
(a) y³/x² (b) y/x (c) y²/x (d) x³/y²
Answer: (a) y³/x²
Solution: Work ∝ men × hours × days. x³ gives x units, so y³ gives y³/x × 1/x = y³/x² units.
Example 8. A group of men decided to finish a work in 15 days. 5 of them did not turn up, so the rest finished it in 30 days. How many men were there originally?
(a) 10 (b) 20 (c) 5 (d) 15
Answer: (a) 10
Solution: M × 15 = (M − 5) × 30 → M = 10.
Example 9. A group of men decided to finish a work in 10 days. 2 of them did not turn up, so the rest finished it in 12 days. How many men were there originally?
(a) 10 (b) 12 (c) 14 (d) 24
Answer: (b) 12
Solution: M × 10 = (M − 2) × 12 → M = 12.
Topic-wise practice for Unitary method, work and wages (Quant): Provisions, wages by work done. Useful for OPSC ASO, OSSC, OSSSC and other Odisha exams.
Question types covered
- Direct and inverse variation
- Provisions lasting for soldiers or students
- Men-days-hours with wages
- Wages divided by work done