Time and work
Treat a whole job as 1. A person who finishes it in d days does 1/d of it each day, so work rates simply add when people work together. Pipes and cisterns follow the same idea, with emptying pipes counted as negative work.
Formulas and when they apply
- Daily work rate = 1 ÷ (days taken alone)Applies when: the worker works at a steady rate
- Together: 1/a + 1/b = 1/T, and for two workers T = ab/(a + b)Applies when: two workers, formula in the second part only
- Work done = rate × time. Time for the rest = remaining fraction ÷ rate
- M₁ × D₁ × H₁ ÷ W₁ = M₂ × D₂ × H₂ ÷ W₂Applies when: men of equal efficiency, D days, H hours per day, W amount of work
- Pipes: a filling pipe has rate +1/t and an emptying pipe −1/t. The net rate is their sum
- Wages are shared in the ratio of work done, which is rate × days worked
- If A is twice as efficient as B, A takes half the time of B for the same job
Worked examples
Example 1. A can do a job in 15 days and B in 10 days. They work together for 4 days and then A leaves. How many more days does B need to finish the job?
Solution. Together they do 1/15 + 1/10 = 2/30 + 3/30 = 1/6 of the job per day. In 4 days they do 4/6 = 2/3 of it, leaving 1/3. B does 1/10 per day, so the rest takes (1/3) ÷ (1/10) = 10/3 = 3⅓ days.
Example 2. 8 men build 120 m of road in 15 days. How many men are needed to build 200 m of road in 10 days at the same rate of work?
Solution. 8 men for 15 days is 120 man-days for 120 m, which is 1 man-day per metre. 200 m needs 200 man-days. In 10 days that needs 200 ÷ 10 = 20 men.
Example 3. Pipe A fills a tank in 6 hours and pipe B empties the full tank in 9 hours. Both are opened together on an empty tank. How long to fill it?
Solution. Net rate = 1/6 − 1/9 = 3/18 − 2/18 = 1/18 of the tank per hour. So the tank fills in 18 hours.
Shortcuts and their limits
- Pick a convenient total work equal to the LCM of the days. If A takes 12 days and B takes 18 days, take the job as 36 units: A does 3 units a day and B does 2.Does not apply when: Only for the proportional work in one job. Do not mix the unit choices of different jobs.
- For two workers, the time together is the product over the sum: 20 and 30 days give 600/50 = 12 days.Does not apply when: Works for two workers only. For three or more, add the rates.
- Efficiency is inversely proportional to the days taken: a 2 : 3 efficiency ratio means a 3 : 2 time ratio.Does not apply when: Needs the same amount of work for both workers.
Common mistakes
- Taking the average of the days instead of adding the daily rates.
- Forgetting to subtract the work already done before finding the days for the rest.
- Treating an emptying pipe as a filling pipe, or leaving it out.
- Counting a worker who leaves early for the whole time.
- Sharing wages by days alone when the workers have different efficiencies.
Independent practice made by EduVistaHQ with original questions. It is not an official examination paper, it is not affiliated with or endorsed by any examination body or company, and a practice score does not predict a real result.