How to Solve Combined Time, Work and Pipes Problems
Solve combined time-work and pipes-and-cisterns problems using additive rates, the LCM trick and outlet subtraction, with worked examples.
Expected Interview Answer
Combined work-rate problems are solved by converting every worker or pipe into a fraction of the job done per unit time, then adding those rates directly — never adding the raw days themselves.
If a worker finishes a job in n days, their rate is 1/n of the job per day; two workers with rates 1/a and 1/b, working together, complete (1/a + 1/b) of the job per day, so the combined time is the reciprocal of that sum. Pipes filling a tank follow the identical mechanic, with an outlet pipe simply contributing a negative rate that subtracts from the total. The trick that collapses fraction arithmetic is assuming a convenient total work size — typically the LCM of all the individual times — so every rate becomes a clean whole number of units per day. Once rates are additive, mixed scenarios such as "A and B work together for 2 days, then A leaves" reduce to simple bookkeeping of units completed versus units remaining.
- Rates add linearly, unlike raw days which never do
- LCM-of-times trick removes fraction arithmetic entirely
- Outlet pipes and part-time leavers are handled as signed rate adjustments
AI Mentor Explanation
If one bowler can bowl through an entire team’s overs alone in 20 overs and a second bowler alone would need 30 overs, their combined over-completion rate is 1/20 + 1/30 of the innings per over, not some average of 20 and 30. Assume the innings is 60 overs worth of "work" (the LCM of 20 and 30): the first bowler clears 3 units per over and the second clears 2 units per over, so together they clear 5 units per over and finish in 12 overs. This unit-rate trick is exactly how combined time-and-work problems, including pipe-filling variants, are solved without ever averaging the individual times.
Worked example (pipe with an outlet)
Pipe A + Pipe B
- 1/20 + 1/30 = 5/60
Minus outlet C
- 5/60 - 1/60 = 4/60
Combined time
- 1 / (4/60) = 15 hours
Step-by-Step Explanation
Step 1
Convert time to rate
Worker/pipe finishing in n units of time has rate 1/n per unit time.
Step 2
Pick a convenient total
Use the LCM of all individual times so every rate is a whole number.
Step 3
Add or subtract rates
Inflow rates add; an outlet pipe or a non-contributing worker subtracts.
Step 4
Invert for combined time
Combined time = total work ÷ combined rate, or 1 ÷ combined fractional rate.
What Interviewer Expects
- Correct conversion of time into a per-unit-time rate
- Recognizing that rates add while raw days never do
- Correct handling of an outlet pipe as a negative rate
- Use of the LCM trick to avoid fraction arithmetic
Common Mistakes
- Averaging the individual days instead of adding rates
- Forgetting to subtract the outlet pipe rate rather than adding it
- Using an inconvenient total work size, leading to fraction errors
- Not adjusting the remaining work when a worker leaves partway through
Best Answer (HR Friendly)
“I convert every worker or pipe into a rate — the fraction of the job it completes per unit time — because rates are what add together, not raw days. I pick a convenient total work size, usually the LCM of the given times, so the rates come out as clean whole numbers. Inflow pipes and additional workers add to the combined rate, while an outlet pipe or someone who stops working subtracts from it, and the final time is just the total work divided by the combined rate.”
Follow-up Questions
- How do you handle a worker who leaves after working for only part of the total time?
- How would you solve for the time taken if two pipes are opened alternately?
- How does efficiency (twice as fast, half as fast) translate into a rate ratio?
- How do you find the time for the outlet alone if the combined and inlet rates are known?
MCQ Practice
1. Pipe A fills a tank in 12 hours and Pipe B fills it in 24 hours. Working together, how long do they take?
Combined rate = 1/12 + 1/24 = 2/24 + 1/24 = 3/24 = 1/8, so combined time = 8 hours.
2. A can do a job in 15 days, B in 10 days. If they work together for 3 days, what fraction of the job remains?
Combined rate = 1/15+1/10 = 1/6 per day; 3 days completes 3/6 = 1/2, leaving 1/2.
3. An inlet pipe fills a tank in 10 hours; an outlet pipe empties it in 15 hours. Both open together, the tank fills in?
Combined rate = 1/10 - 1/15 = 3/30 - 2/30 = 1/30, so it takes 30 hours.
Flash Cards
How to convert time to a rate? — Rate = 1 / time taken to finish the whole job alone.
How do combined rates behave? — They add for inflows; an outlet pipe subtracts its rate.
Trick to avoid fractions? — Set total work = LCM of all the given individual times.
Formula for combined time? — Combined time = total work ÷ combined rate (or 1 ÷ sum of rates).