Solve problems involving pipes filling tanks, people completing jobs, and combined work rates
Explanation & Worked Examples
The Core Idea: Rate = 1 ÷ Time
Every worker or pipe has a rate — the fraction of the job it completes per unit of time.
Rate = 1 ÷ Time to complete the whole job
For example:
Pipe A fills a tank in 4 hours → Rate = 1/4 of the tank per hour
Person B completes a job in 10 days → Rate = 1/10 of the job per day
Why Rates Add
When two or more workers/pipes operate at the same time, their rates add together because each is contributing independently:
Combined Rate = Rate₁ + Rate₂ + Rate₃ …
A drain pipe subtracts from the combined rate instead of adding.
Think of it this way: If Pipe A fills 1/4 of the tank every hour and Pipe B fills 1/6 every hour, after one hour together they have filled 1/4 + 1/6 = 5/12 of the tank. The whole job takes 12/5 hours.
The Formula
To find the time taken when multiple sources work together:
Step 1 — Find each rate
Rate = 1 ÷ (individual time)
Step 2 — Add (or subtract) rates
Combined Rate = R₁ + R₂ − R_drain
Step 3 — Find combined time
Time = 1 ÷ Combined Rate
Shortcut for Two Sources
When exactly two pipes/workers are involved, there is a direct formula:
T = (A × B) ÷ (A + B)
where A and B are the individual times. This only works for two filling sources — do not use it when there is also a drain.
Common mistake: Do not average the two times. If Pipe A takes 4 h and Pipe B takes 6 h, the answer is NOT (4 + 6) / 2 = 5 h. It is (4 × 6) / (4 + 6) = 24/10 = 2.4 h — faster than either pipe alone.
Easy Examples
Direct application of the combined rate formula.
Example 1 — Two Pipes
Pipe A fills a tank in 4 hours. Pipe B fills it in 6 hours. Both are opened together. How long to fill the tank?
A and B together finish a job in 6 days. A alone takes 10 days. How long does B alone take?
1Combined rate = 1/6 | Rate of A = 1/10
2Rate of B = 1/6 − 1/10 = 5/30 − 3/30 = 2/30 = 1/15
3B alone = 15 days
Example 3 — Delayed Start
Pipe A fills a tank in 6 hours. Pipe B fills it in 4 hours. A is opened first. After 2 hours, B is also opened. How long in total to fill the tank?
1Work done by A in 2 h = 2 × 1/6 = 1/3 of the tank
2Remaining work = 1 − 1/3 = 2/3
3Combined rate (A + B) = 1/6 + 1/4 = 5/12 per hour
4Time for remaining 2/3 = (2/3) ÷ (5/12) = (2/3) × (12/5) = 8/5 = 1.6 h
5Total time = 2 + 1.6 = 3 hours 36 minutes
Drain pipe rule: Only subtract the drain rate if the drain is open at the same time as the fill pipe. If the tank is being drained after filling, treat them as separate stages.
Complex Examples
Three sources, partial work, or working backwards from the result.
Example 1 — Three Pipes (Two Fill, One Drains)
Pipe A fills in 3 hours, Pipe B fills in 4 hours, Pipe C drains in 6 hours. All three are open. How long to fill the tank?
Key pattern: If working together is faster than either alone (which it always should be for filling), your combined rate is correct. If you get a combined time longer than either individual time, you have made a sign error — likely subtracted instead of added, or vice versa.