Work Rate & the LCM Method

🎯 Learning Objective

Model work as rate per day and combine workers by adding rates using the LCM trick.

💡 Concept

• A finishes a job in n days → A does 1/n of the work per day
• Working together → RATES add, days never do
• LCM method: let total work = LCM of the day-counts, so every rate becomes whole units/day
• Time = Total work ÷ Combined rate

🧮 Key Formulas

1 day's work = 1/n

>

Together: 1/a + 1/b

>

Time = Work ÷ Rate

✏️ Easy Example

Q. A can do a job in 10 days and B in 15 days. In how many days will they finish it together?

1. Work = LCM(10, 15) = 30 units
2. A = 3/day, B = 2/day → together 5/day
3. 30 ÷ 5 = 6

Answer: 6 days

🇮🇳 Real-Life Example

Diwali ki safai: alone you need 6 hours, your sister also needs 6 — together the house sparkles in 3. Rates added, time halved.

📝 Exam-Level Example

Q. A and B together can do a work in 12 days; A alone can do it in 20 days. In how many days can B alone do it?

1. Work = LCM(12, 20) = 60 units
2. Together = 5/day, A = 3/day
3. B = 2/day → 60 ÷ 2 = 30

Answer: 30 days

📝 Exam-Level Example

Q. A, B and C can do a work in 12, 15 and 20 days respectively. In how many days will they finish it together?

1. Work = LCM = 60 units
2. Rates: 5 + 4 + 3 = 12/day
3. 60 ÷ 12 = 5

Answer: 5 days

🪄 Memory Trick

Assume work = LCM of all the day-counts. Whole-number rates, zero fractions, half the solving time.

⚠️ Common Mistakes

• ❌ Adding days instead of rates (10 + 15 = 25 days is wrong)
• ❌ Forgetting to convert the combined rate back into time at the end

🏆 Exam Tips

• ✅ Combined time is always LESS than the fastest individual time — instant sanity check
• ✅ Write each worker's units/day above their name before touching the question

📌 Summary

• Rate = 1/n per day
• Together → add rates
• LCM method kills fractions
• Time = work ÷ combined rate