About Time and Work
Time and work questions are about rates, not about time. A worker who finishes a job in 12 days completes one twelfth of it each day, and that fraction is what you add and subtract. Once you stop thinking in days and start thinking in "work per day", the whole topic becomes a single addition of fractions.
What you need to understand
- Work is measured as a fraction of the whole job. If A finishes a job in a days, A does 1/a of it in one day.
- Rates add when people work together: the combined rate is the sum of the individual rates.
- Times do not add. Two workers who each need 10 days do not need 20 days together, they need 5.
- If A is n times as efficient as B, A takes 1/n of the time B takes.
- Work done is rate multiplied by time, so the fraction completed in d days is d divided by the total time.
- Work left is always one minus the work completed, and the remaining work is finished at the rate of whoever is still working.
Formulas to remember
How to work through these questions
- Convert every time into a rate: 1/a, 1/b and so on. Write the rates down before doing anything else.
- Add the rates of everyone working at the same time.
- Convert the total rate back into a time by taking its reciprocal.
- For questions where someone leaves or joins partway, find the work completed in the first period, subtract it from one, then divide the remainder by the rate of whoever is left.
- Check that the answer is smaller than the fastest individual time, and never larger than the slowest.
Mistakes that cost marks
- Adding the times instead of the rates. Ten days and ten days is five days together, not twenty.
- Taking the average of two times, which is wrong for the same reason.
- Forgetting that part of the work is already finished before someone leaves.
- Mixing units — days with hours, or workers with hours.
- Answering with the time of the wrong person when the question asks who finishes alone.
Worked example
A can finish a piece of work in 12 days and B can finish the same work in 6 days. Working together, how many days will they take?
- A completes 1/12 of the work in one day and B completes 1/6 of the work in one day.
- Working together they complete 1/12 + 1/6 = 1/12 + 2/12 = 3/12 = 1/4 of the work per day.
- If one quarter of the work is finished each day, the whole job takes 4 days.
Answer: 4 days
Practice questions with answers
A few Time and Work questions with the full solution shown, so you can see
how the method is applied before you attempt the timed set.
Question 1
A can finish a piece of work in 14 days and B can finish the same work in 35 days. Working together, in how many days will they finish the work?
Answer: Option A — with explanation
Work done by A in one day = 1/14 and work done by B in one day = 1/35.
Working together they finish 1/14 + 1/35 = 1/10 of the work per day.
Total time = (14 x 35) / (14 + 35) = 10 days.
Common mistakes
- took the average of the two times: 24.5
- added the two times instead of adding the work rates: 49
- subtracted the two times instead of combining the rates: 21
Question 2
A and B finish a task together in 60 days, and A alone takes 108 days. Find the time the other person takes working alone.
Answer: Option B — with explanation
Work done together in one day = 1/60 and the known worker does 1/108 of it per day.
So the other worker does 1/60 - 1/108 = 1/135 of the work per day.
Time taken alone = 135 days.
Common mistakes
- applied the together-time formula again instead of rearranging it: 38.57
- subtracted the together time from the known worker's time: 48
- added the two times instead of subtracting the rates: 168
Question 3
A can finish a piece of work in 14 days and B can finish the same work in 35 days. They work together for 6 days, after which A leaves. In how many more days will B finish the remaining work?
Answer: Option A — with explanation
Working together they finish 1/14 + 1/35 = 1/10 of the work per day.
In 6 days they finish 6/10 of the work, so 1 - 6/10 = 4/10 of the work is left.
B alone does 1/35 per day, so the remaining work takes 35 x 4/10 = 14 days.
Common mistakes
- used the work already finished instead of the work remaining: 21
- subtracted the days already worked from B's total time: 29
- confused the fraction of work left with a number of days: 4
Question 4
A can finish a piece of work in 60 days and B can finish the same work in 90 days. Working together, in how many days will they finish the work?
Answer: Option A — with explanation
Work done by A in one day = 1/60 and work done by B in one day = 1/90.
Working together they finish 1/60 + 1/90 = 1/36 of the work per day.
Total time = (60 x 90) / (60 + 90) = 36 days.
Common mistakes
- subtracted the two times instead of combining the rates: 30
- took the average of the two times: 75
- added the two times instead of adding the work rates: 150
Question 5
A and B together can finish a piece of work in 14 days. B alone can finish it in 63 days. In how many days can the other person alone finish it?
Answer: Option C — with explanation
Work done together in one day = 1/14 and the known worker does 1/63 of it per day.
So the other worker does 1/14 - 1/63 = 1/18 of the work per day.
Time taken alone = 18 days.
Common mistakes
- added the two times instead of subtracting the rates: 77
- subtracted the together time from the known worker's time: 49
- applied the together-time formula again instead of rearranging it: 11.45
Frequently asked questions
Why can we not just add the number of days?
Because the two workers are working at the same time. Adding days would mean one worker starts only after the other has finished. What adds is the work done per day, not the time taken.
What is the formula when two people work together?
If A alone takes a days and B alone takes b days, together they take ab/(a + b) days. This comes directly from adding the rates 1/a and 1/b and then taking the reciprocal.
How do I handle a question where one worker leaves in the middle?
Split the job into phases. Work out how much was completed while both were working, subtract it from one to get the remainder, then divide that remainder by the rate of the worker who stays.
What does it mean if A is twice as efficient as B?
A does twice as much work in the same time, so A needs half the time B needs. If B takes 30 days, A takes 15.
Are pipes and cistern questions the same topic?
The mathematics is identical — filling pipes add rates and emptying pipes subtract them. They are usually listed as a separate topic, but if you can do Time and Work you can already do Pipes and Cistern.
