Two methods
Fraction method: if A finishes in a days, A does 1/a of the work per day. Efficiency (LCM) method, used by toppers: take total work = LCM of the days; each person's efficiency = total/days. This avoids fractions completely. Target: 45 seconds.
Efficiency method example
A takes 12 days, B takes 18 days. Total work = LCM(12, 18) = 36 units. A = 3 units/day, B = 2 units/day. Together = 5 units/day, time = 36/5 = 7.2 days.
Key formulas
| Situation | Formula |
|---|---|
| A and B together | ab/(a + b) days |
| A, B, C together | abc/(ab + bc + ca) days |
| A + B, B + C, C + A given | 2(A + B + C) = sum of the three pair rates |
| Men-days-hours | M1 D1 H1 / W1 = M2 D2 H2 / W2 |
| A is k times as efficient as B | time ratio A : B = 1 : k |
| Wages | shared in the ratio of work done (efficiency x days) |
Leaving and joining
- "A leaves after 4 days": A contributes 4 x efficiency units.
- "B leaves 3 days before completion": B works (T - 3) days; set up the equation in T.
- "Alternate days": work in 2-day cycles, then handle the last part.
Worked example
10 men working 8 hours a day finish a job in 12 days. How many days will 15 men working 8 hours a day take?
10 x 12 x 8 = 15 x D x 8, so D = 8 days.
Worked example 2
A and B can do a work in 15 and 20 days. With C they finish in 6 days and earn Rs. 4,500. C's share?
Total = 60 units: A = 4, B = 3, all three = 10, so C = 3. Shares 4 : 3 : 3, C = Rs. 1,350.
Common traps
- Adding days instead of rates (12 + 18 days is not the combined time).
- Forgetting that wages depend on work done, not on days worked alone.
- In "B leaves 3 days before completion" questions, the unknown is the total time, not B's time.
- Double work means W2 = 2 in the MDH formula.
Worked example 3 (pair rates)
A + B take 12 days, B + C take 15 days and C + A take 20 days. Total work = 60 units, so the pair rates are 5, 4 and 3 units/day. 2(A + B + C) = 12, so all three together do 6 units/day and finish in 10 days. A alone = 6 - 4 = 2 units/day, i.e. 30 days.