Time & Work • Topic 2 of 5

Combined Work

When several people work together, their efficiencies simply add. Convert each person’s days into units/day using total work = LCM of the days, sum the rates, and divide the total work by that sum. For the common two-worker case there is the well-known shortcut: if A takes a days and B takes b days, together they take ab/(a+b) days — but the LCM method generalises to any number of workers and to leaving/joining problems without memorising a new formula. The same machinery handles "A and B work for some days, then A leaves": track units completed, subtract from the total, and divide the remainder by whoever is still working. A worker who hinders the job (a leak, a return of goods, an opposing pump) contributes a negative rate, which you subtract. Keeping everything in whole units is what keeps mid-problem arithmetic clean under exam pressure.

✅ Solved examples

1. A does a job in 12 days, B in 18 days. Working together, how many days to finish?
Total work = LCM(12,18) = 36. A = 3, B = 2 ⇒ together 5 units/day ⇒ 36/5 = 7.2 days.
2. A, B, C take 10, 12, 15 days respectively. Together they finish in?
Total work = LCM(10,12,15) = 60. Rates 6 + 5 + 4 = 15 units/day ⇒ 60/15 = 4 days.
3. A and B together do a job in 4 days; A alone in 12 days. They earn ₹6,000. Find B’s share.
Total work = LCM(4,12) = 12. Together 3/day, A = 1/day ⇒ B = 2/day. Wages split as efficiency = 1 : 2 ⇒ B gets (2/3) × 6000 = ₹4,000.
4. A starts a job alone (15 days total). After 3 days B joins (B alone needs 10 days). When is it finished?
Total work = LCM(15,10) = 30. A = 2/day, B = 3/day. A’s 3 days = 6 units; 24 left at 5/day = 4.8 days ⇒ total 7.8 days.

✏️ Practice — try these, take hints as needed

1. A finishes in 6 days, B in 8 days. Together they take how long?
Use ab/(a+b).
(6×8)/(6+8).
48/14.
24/7 ≈ 3.43 days
2. A, B, C take 20, 30, 60 days. Together they finish in?
Total work = LCM = 60.
Rates 3 + 2 + 1.
60 ÷ 6.
10 days
3. A and B together: 12 days; B and C together: 15 days; A and C: 20 days. All three together?
Total work = LCM(12,15,20) = 60; pair rates 5, 4, 3.
Sum of pairs = 2(A+B+C) = 12 ⇒ A+B+C = 6/day.
60 ÷ 6.
10 days
4. A alone needs 24 days. A and B together for 6 days do half the job. B alone needs?
Total work = 24 units; half = 12 units in 6 days ⇒ combined 2/day.
A = 1/day ⇒ B = 1/day.
24 ÷ 1.
24 days
5. A and B do a job in 10 days for ₹4,500. A alone takes 15 days. B’s wage share?
Total work = LCM(10,15) = 30; together 3/day, A = 2/day.
B = 1/day ⇒ ratio A : B = 2 : 1.
B gets 1/3 of 4500.
₹1,500

📝 Topic test — 8 questions

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