Wages — Payment in Ratio of Work

🎯 Learning Objective

Split wages in proportion to work done, which equals rate × days worked.

💡 Concept

• Wages are shared in the ratio of WORK DONE, not days present
• Same number of days worked → share ratio = rate ratio (1/a : 1/b)
• Share = (person's units ÷ total units) × total money
• A helper who joins to speed things up is paid only for the gap he actually fills

🧮 Key Formulas

Share ∝ work done = rate × days

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Equal days ⇒ shares in ratio 1/a : 1/b

✏️ Easy Example

Q. A and B can do a job in 6 and 12 days. Working together they earn ₹900. Find A's share.

1. Work = LCM(6, 12) = 12 → A = 2/day, B = 1/day
2. Ratio A : B = 2 : 1
3. A = (2/3) × 900

Answer: ₹600

🇮🇳 Real-Life Example

Two painters finish your flat together — the faster one covers two walls to the other's one. When cash is handed over they never split 50-50; the ustad rightfully takes double.

📝 Exam-Level Example

Q. A, B and C can do a work in 6, 8 and 12 days. They work together and are paid ₹1,350. Find B's share.

1. Work = LCM = 24 → rates 4, 3, 2
2. Ratio = 4 : 3 : 2, total 9 parts
3. B = (3/9) × 1350

Answer: ₹450

📝 Exam-Level Example

Q. A does a job in 10 days, B in 15 days. With the help of C they finish in 5 days. If the total payment is ₹450, find C's share.

1. Work = 30 → A = 3, B = 2, needed = 30/5 = 6/day
2. C = 6 − 5 = 1/day
3. C = (1/6) × 450

Answer: ₹75

🪄 Memory Trick

Wages ratio = units ratio. Compute LCM units once and the money divides itself.

⚠️ Common Mistakes

• ❌ Splitting money by days present instead of work done
• ❌ Forgetting C's rate = required combined rate − (A + B)

🏆 Exam Tips

• ✅ Same days for all → shares follow efficiency directly
• ✅ Different days → multiply each rate by own days first, then take the ratio

📌 Summary

• Payment follows work done
• Equal days → efficiency ratio
• Helper's rate = needed − existing
• Share = own units ÷ total units × money