60-second revision
- Take total work = LCM of the given days; each person's rate = total ÷ own days.
- A in a days, B in b days: together ab/(a + b) days.
- Twice as efficient means half the time.
- M1 × D1 × H1 / W1 = M2 × D2 × H2 / W2.
- Pairs take x, y, z days: all three take 2xyz/(xy + yz + zx) days.
- An outlet or leak is negative work; fill time with a leak gives leak time = ab/(b − a).
- Wages are split in the ratio of work done.
Think in units, not fractions
Instead of adding fractions like 1/10 + 1/15, take the total work as the LCM of the days. Each worker's efficiency is then a whole number of units per day.
Worked example: A finishes a job in 10 days, B in 15 days. Total = 30 units. A does 3 units a day, B does 2, together 5. They finish in 30/5 = 6 days.
| Given | Result |
|---|---|
| A alone a days, B alone b days | Together ab/(a + b) days |
| Together t days, A alone a days | B alone at/(a − t) days |
| A+B x days, B+C y days, C+A z days | All three 2xyz/(xy + yz + zx) days |
| A, B, C alone a, b, c days | Together abc/(ab + bc + ca) days |
Worked example on pairs: A and B take 10 days, B and C take 12, C and A take 15. All three = 2 × 1800/(120 + 180 + 150) = 3600/450 = 8 days.
Efficiency
Efficiency is work per day, and it is inversely proportional to time. If A is three times as efficient as B, A takes one-third of B's time.
Worked example: A is thrice as efficient as B and together they finish in 9 days. Rates 3 + 1 = 4 units a day, so work = 36 units. A alone: 12 days; B alone: 36 days.
Men, days and hours
When more people, days or hours are involved, use M1 × D1 × H1 / W1 = M2 × D2 × H2 / W2, where W is the amount of work.
Worked example: 6 men working 8 hours a day dig a 40 m trench in 10 days. How long will 8 men working 6 hours a day take to dig 50 m? 6 × 10 × 8/40 = 8 × D × 6/50, so 12 = 48D/50 and D = 12.5 days.
Someone leaves or joins
Work out how much is done while everyone works, then give the remainder to whoever is left.
Worked example: A takes 20 days and B 30 days. Total 60 units; A does 3 a day, B 2. Together for 6 days they finish 30 units. B then leaves; A finishes the other 30 units in 10 days, so the job takes 16 days in all.
Pipes and cisterns
Pipes are workers. An inlet does positive work, an outlet or leak does negative work. Use the same LCM-units method.
Worked example: pipe A fills a tank in 12 hours, B in 15 hours, and C empties it in 20 hours. Total 60 units: +5, +4, −3, net 6 units an hour. All three together fill it in 10 hours.
| Situation | Formula |
|---|---|
| Inlet fills in a h, outlet empties in b h (b > a) | Both open: fills in ab/(b − a) h |
| Fills in a h normally, b h with a leak | Leak alone empties in ab/(b − a) h |
Sharing wages
Pay is shared in the ratio of work done. If everyone works the same number of days, that is simply the ratio of efficiencies, which is the inverse ratio of their individual days.
Worked example: A can do a job in 12 days and B in 20 days. Efficiency ratio = 1/12 : 1/20 = 5:3. Out of Rs 4,000, A gets Rs 2,500 and B gets Rs 1,500.
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