Time and work
If A finishes a job in n days, A does 1/n of it each day. Add daily work to combine people; the LCM method avoids fractions.
Formula sheet
| Situation | Formula |
|---|---|
| A does a job in a days | A's one day's work = 1/a |
| A and B together | Time = ab ÷ (a + b) |
| A, B and C together | Time = abc ÷ (ab + bc + ca) |
| A and B together take t days; A alone a days | B alone = at ÷ (a − t) |
| LCM (units) method | Total work = LCM of the days; each person's units per day = total ÷ own days |
| Men, days, hours and work | M₁ × D₁ × H₁ ÷ W₁ = M₂ × D₂ × H₂ ÷ W₂ |
| Wages | Shared in the ratio of work done |
| Pipes and cisterns | Filling rates add; emptying rates subtract |
Worked example 1: two people together
Lata can stitch a batch of uniforms in 12 days and Meena can stitch it in 18 days. How long will they take working together?
- Standard method (fractions): Lata does 1/12 a day and Meena 1/18 a day. Together: 1/12 + 1/18 = 3/36 + 2/36 = 5/36 a day. Time = 36/5 = 7.2 days.
- Shortcut (LCM units): total work = LCM(12, 18) = 36 units. Lata does 3 units a day and Meena 2 units, so together 5 units a day. Time = 36 ÷ 5 = 7.2 days. (Or ab ÷ (a + b) = 216 ÷ 30 = 7.2.)
- Check: in 7.2 days Lata does 7.2/12 = 0.6 of the work and Meena does 7.2/18 = 0.4; together 1 whole job.
7.2 days (7 1/5 days).
Worked example 2: one person leaves midway
A can do a piece of work in 20 days and B in 30 days. They work together for 6 days, and then A leaves. In how many more days will B finish the work?
- Standard method: together they do 1/20 + 1/30 = 5/60 = 1/12 a day. In 6 days: 6/12 = 1/2 of the work. Remaining 1/2 at B's rate of 1/30 a day takes (1/2) ÷ (1/30) = 15 days.
- Shortcut (LCM units): work = 60 units; A does 3 a day, B 2 a day, together 5. In 6 days: 30 units. Remaining 30 units at 2 a day = 15 days.
- Check: B works 6 + 15 = 21 days = 42 units; A works 6 days = 18 units; 42 + 18 = 60 units.
15 more days.
Worked example 3: men, days and hours
12 workers, working 8 hours a day, finish a job in 10 days. How many workers, working 6 hours a day, are needed to finish twice the job in 16 days?
- Standard method (worker-hours): one job = 12 × 8 × 10 = 960 worker-hours, so two jobs = 1,920 worker-hours. Each new worker gives 6 × 16 = 96 hours. Workers needed = 1,920 ÷ 96 = 20.
- Shortcut (formula): M₁D₁H₁ ÷ W₁ = M₂D₂H₂ ÷ W₂, so 12 × 10 × 8 ÷ 1 = M₂ × 16 × 6 ÷ 2. M₂ = 12 × 10 × 8 × 2 ÷ (16 × 6) = 20.
- Check: 20 × 6 × 16 = 1,920 worker-hours = 2 × 960.
20 workers.
Worked example 4: pipes and a leak
Pipe A fills a tank in 10 hours and pipe B in 15 hours. A leak empties the full tank in 30 hours. With all three working, how long does the tank take to fill?
- Standard method: net rate = 1/10 + 1/15 − 1/30 = 3/30 + 2/30 − 1/30 = 4/30 = 2/15 of the tank an hour. Time = 15/2 = 7.5 hours.
- Shortcut (LCM units): tank = 30 units; A adds 3, B adds 2 and the leak removes 1 each hour, net 4 units. Time = 30 ÷ 4 = 7.5 hours.
- Check: in 7.5 hours A adds 22.5, B adds 15 and the leak removes 7.5 units: 22.5 + 15 − 7.5 = 30 units.
7.5 hours (7 hours 30 minutes).
Bank, SSC and railway papers ask two or three people together, 'one leaves midway', men-days-hours and pipes and cisterns; placement tests add wage sharing. The LCM 'units of work' method avoids fractions and saves time.
2 marks: “A can do a piece of work in 10 days and B in 15 days. In how many days can they do it together?” ab ÷ (a + b) = 10 × 15 ÷ 25 = 6 days.
5 marks: “A can do a work in 20 days and B in 30 days. They work together for 6 days, and then A leaves. In how many more days will B finish it?” Together 1/12 a day; 6 days finish 1/2; B finishes the other 1/2 in 15 days. Check with 60 units.
10 marks: “(a) Lata takes 12 days and Meena 18 days: find their time together. (b) 12 workers at 8 hours a day take 10 days: how many workers at 6 hours a day finish twice the job in 16 days? (c) Pipes fill a tank in 10 and 15 hours and a leak empties it in 30 hours: find the filling time.” (a) 7.2 days. (b) 20 workers. (c) 7.5 hours. Use the LCM method and add a check line to each.
Work = rate × time. Take total work = LCM of the days, find units per day, then divide. M₁D₁H₁/W₁ = M₂D₂H₂/W₂.