How to Solve Mixed Fill-and-Empty Pipe Problems Opened in Stages
Solve staged pipe fill-and-empty aptitude problems using segment-by-segment rate tracking, with a worked example and practice.
Expected Interview Answer
Break the timeline into stages based on when each pipe is opened or closed, compute the net signed rate active during each stage, find how much of the tank fills or empties during that stage's known duration, then solve the final stage for the remaining unknown time.
Whenever pipes are opened or closed partway through a problem, split the total time into segments where the active set of pipes (and hence the net rate) stays constant. For each completed segment, multiply its net rate by its duration to get the fraction of the tank filled or emptied during that segment, and track a running total of the tank's state. The final segment usually has the unknown duration: set up an equation where the running total plus the final segment's rate times its unknown duration equals 1 (full) or 0 (empty), then solve for that duration. This stage-by-stage bookkeeping is what separates these problems from simple constant-pipe problems, and skipping it is the most common source of error.
- Segmenting by pipe changes prevents applying the wrong rate to the wrong interval
- Running-total tracking makes the final unknown-time equation straightforward
- Generalizes to any number of open/close events, not just one switch
AI Mentor Explanation
A team scores at 6 runs per over for the first 10 overs, then a rain-enforced faster required rate of 9 runs per over kicks in for the remainder — you cannot average 6 and 9 across all overs, you must compute the first 10-over segment's total (60 runs) separately, then solve how many more overs at 9 runs per over are needed to reach the target from there. Mixed fill-and-empty pipe problems with staged pipe changes use this identical segment-by-segment approach: compute each constant-rate stage's contribution before solving the final unknown stage.
Worked example (pipe added partway through)
Stage 1 (A alone, 4 hrs)
- 4 × 1/12 = 1/3 filled
Stage 2 (A+B)
- Rate = 5/24
- Remaining 2/3 → 3.2 hrs
Total time
- 4 + 3.2 = 7.2 hours
Step-by-Step Explanation
Step 1
Identify the stage boundaries
Mark every point where a pipe is opened or closed.
Step 2
Compute each completed stage
Net rate × known duration = fraction filled/emptied in that stage.
Step 3
Track the running total
Sum completed stages' contributions before moving to the next.
Step 4
Solve the final stage
Set running total + final rate × unknown time = 1 (or 0), solve for time.
What Interviewer Expects
- Correctly identifies every point where the active rate changes
- Computes each stage's contribution before moving to the next
- Sets up the correct equation for the unknown final stage
- Does not apply a single blended rate across the whole timeline
Common Mistakes
- Averaging rates across stages instead of computing each segment separately
- Forgetting to subtract the completed stages' progress before solving the last stage
- Misidentifying which pipes are active during a given stage
- Setting up the final equation against the wrong target (0 vs 1)
Best Answer (HR Friendly)
“I break the timeline at every point a pipe turns on or off, work out how much of the tank each constant-rate stage fills or empties using its known duration, and keep a running total. For the final stage, I set up an equation where the running total plus the new rate times the unknown time equals a full or empty tank, and solve for that time.”
Follow-up Questions
- How would you handle a problem with three separate pipe-opening events?
- What changes if a pipe is closed rather than opened partway through?
- How do you set up the equation if the tank starts partially full?
- How would you verify your staged-rate answer using a sanity check?
MCQ Practice
1. Pipe A fills in 10 hours alone. It runs for 3 hours, then Pipe B (fills in 15 hours alone) joins. How long after B joins does the tank finish filling?
Stage 1: 3×(1/10)=3/10 filled. Remaining 7/10. Combined rate = 1/10+1/15 = 1/6. Time = (7/10)/(1/6) = 4.2 hours (answer B corrected: 4.2 hours is the accurate result).
2. In staged pipe problems, why can't you use one averaged rate for the whole timeline?
The net rate is only constant within a stage where the same pipes are active; it changes whenever a pipe opens or closes.
3. A tank is 1/4 full when Pipe C (empties in 6 hours) is opened alone. How long to empty the remaining tank?
Remaining fraction to empty = 1/4 of the tank; empty rate = 1/6 per hour; time = (1/4)/(1/6) = 1.5 hours.
Flash Cards
What triggers a new “stage” in a mixed pipe problem? — Any point where a pipe is opened or closed, changing the active net rate.
How do you compute a completed stage's contribution? — Net rate for that stage × its known duration.
How do you find the unknown final duration? — Solve: running total + final rate × unknown time = 1 (full) or 0 (empty).
Biggest pitfall in staged pipe problems? — Using one blended average rate instead of computing each stage separately.