Chain Rule — M·D·H / W
Chain rule — M D H का formula
Chain Rule — M·D·H / W
- Time & Work
- Chain Rule — M·D·H / W
Use M₁D₁H₁/W₁ = M₂D₂H₂/W₂ to handle changing men, days, hours and work.
🎯 Learning Objective
Use M₁D₁H₁/W₁ = M₂D₂H₂/W₂ to handle changing men, days, hours and work.
💡 Concept
- Total effort = Men × Days × Hours; equal work means equal effort
- M₁D₁H₁/W₁ = M₂D₂H₂/W₂ — strike out any factor that does not change
- More men → fewer days (inverse); more work → more days (direct)
- '3 men = 6 boys' type statements: convert the whole workforce into ONE unit first
🧮 Key Formulas
M₁D₁H₁/W₁ = M₂D₂H₂/W₂
✏️ Easy Example
Q. 12 men can finish a work in 10 days. In how many days will 15 men finish it?
- M × D constant: 12 × 10 = 15 × D
- D = 120 ÷ 15
Answer: 8 days
🇮🇳 Real-Life Example
Metro construction before an inauguration: the contractor doubles the labour gang to halve the remaining days — every site engineer runs M × D × H in his head.
📝 Exam-Level Example
Q. 6 men working 8 hours a day finish a job in 10 days. In how many days will 8 men working 6 hours a day finish it?
- 6 × 8 × 10 = 8 × 6 × D
- 480 = 48 × D
- D = 10
Answer: 10 days
📝 Exam-Level Example
Q. 10 men can build a 100 m wall in 5 days. How many men are needed to build a 60 m wall in 3 days?
- (10 × 5)/100 = (M × 3)/60
- 1/2 = M/20
- M = 10
Answer: 10 men
🪄 Memory Trick
Cancel every unchanged quantity from both sides first — what survives is usually one multiplication and one division.
⚠️ Common Mistakes
- ❌ Putting work W on the wrong side — W divides, it never multiplies
- ❌ Treating men and days as directly proportional (they are inverse)
🏆 Exam Tips
- ✅ Logic check at the end: fewer hours or harder work must push days UP
- ✅ For men-boys conversions, express everyone in boys (or men) before using the formula
📌 Summary
- Effort = M × D × H
- M₁D₁H₁/W₁ = M₂D₂H₂/W₂
- Men-days inverse, work-days direct
- Cancel the unchanged, then solve