How to Solve: Pipes and Cisterns Problems

How to Solve: Pipes and Cisterns Problems

(Complete Guide for SSC/Bank/Railway Exams)

Introduction

"Mastering Pipes and Cisterns problems can add 3–5 marks to your SSC, Bank, or Railway exam score—enough to push you into the next cutoff bracket. These questions test your ability to handle work rates, just like filling a tank at home, but with exam-level twists."

What You Need To Know First

1. Work-Rate Concept: If a person completes a job in n days, their work rate is 1/n of the job per day.
2. LCM Method: Used to find a common base for work rates (e.g., if Pipe A fills in 6 hours and Pipe B in 4 hours, LCM of 6 and 4 is 12).
3. Positive/Negative Work: Filling pipes = positive work; emptying pipes = negative work.

Key Vocabulary

Formulas To Know

1. Single Pipe Rate
2. Formula: Rate = 1 / Time
3. Time = Time taken to fill/empty the cistern alone.

4. MEMORISE THIS.

5. Combined Rate (Multiple Pipes)

6. Formula: Total Rate = Rate₁ + Rate₂ + ... ± Rateₙ
7. Use + for filling pipes, – for emptying pipes.

8. MEMORISE THIS.

9. Time to Fill/Empty

10. Formula: Time = 1 / Total Rate
11. Total Rate = Combined rate of all pipes.

12. MEMORISE THIS.

13. LCM Method (For Efficiency)

14. Step 1: Find LCM of individual times.
15. Step 2: Assume LCM as total work (e.g., 12 units).
16. Step 3: Calculate rates in units/hour.
17. Given on exam sheet (but practice this!).

Step-by-Step Method

Step 1: Identify Pipes and Their Rates

• List all pipes and their individual times to fill/empty the cistern.
• Convert times to rates using Rate = 1 / Time.

Step 2: Determine if Pipes Fill or Empty

• Filling pipes = + rate.
• Emptying pipes (leaks, outlet pipes) = – rate.

Step 3: Calculate Combined Rate

• Add rates of all pipes (use + for filling, – for emptying).
• Example: Pipe A (1/5) + Pipe B (1/10) – Leak (1/20) = (4/20 + 2/20 – 1/20) = 5/20 = 1/4.

Step 4: Find Total Time

• Use Time = 1 / Total Rate.
• Example: Total Rate = 1/4 → Time = 4 hours.

Step 5: Adjust for Partial Filling (If Needed)

• If the question asks for time to fill part of the cistern (e.g., 3/4 full), multiply total time by the fraction.
• Example: Time to fill 3/4 = 4 × (3/4) = 3 hours.

Worked Examples

Example 1 – Basic

Question: Pipe A fills a cistern in 6 hours. Pipe B fills it in 4 hours. How long will it take to fill the cistern if both pipes are opened together?

Solution: 1. Pipe A’s rate = 1/6 cistern/hour. 2. Pipe B’s rate = 1/4 cistern/hour. 3. Combined rate = 1/6 + 1/4 = (2 + 3)/12 = 5/12 cistern/hour. 4. Time to fill = 1 / (5/12) = 12/5 = 2.4 hours = 2 hours 24 minutes.

What we did and why: - Converted individual times to rates. - Added rates (both filling). - Used Time = 1 / Total Rate to find the answer.

Example 2 – Medium (With Leak)

Question: Pipe A fills a cistern in 3 hours. Pipe B fills it in 6 hours. A leak empties the cistern in 12 hours. If all three are opened together, how long will it take to fill the cistern?

Solution: 1. Pipe A’s rate = 1/3 cistern/hour. 2. Pipe B’s rate = 1/6 cistern/hour. 3. Leak’s rate = –1/12 cistern/hour (negative because it empties). 4. Combined rate = 1/3 + 1/6 – 1/12 = (4/12 + 2/12 – 1/12) = 5/12 cistern/hour. 5. Time to fill = 1 / (5/12) = 12/5 = 2.4 hours = 2 hours 24 minutes.

What we did and why: - Treated the leak as a negative rate. - Added all rates (filling + emptying). - Calculated time using the combined rate.

Example 3 – Exam-Style (Disguised)

Question: A tank can be filled by Pipe X in 8 hours and by Pipe Y in 12 hours. Pipe Z can empty the tank in 24 hours. If Pipe X is opened at 9 AM, Pipe Y at 10 AM, and Pipe Z at 11 AM, at what time will the tank be full?

Solution: 1. Pipe X’s rate = 1/8 cistern/hour. 2. Pipe Y’s rate = 1/12 cistern/hour. 3. Pipe Z’s rate = –1/24 cistern/hour. 4. From 9 AM to 10 AM (1 hour): Only Pipe X is open.
- Work done = 1 × (1/8) = 1/8 cistern. 5. From 10 AM to 11 AM (1 hour): Pipes X and Y are open.
- Combined rate = 1/8 + 1/12 = 5/24 cistern/hour.
- Work done = 1 × (5/24) = 5/24 cistern.
- Total work so far = 1/8 + 5/24 = 3/24 + 5/24 = 8/24 = 1/3 cistern. 6. From 11 AM onwards: All three pipes are open.
- Combined rate = 1/8 + 1/12 – 1/24 = (3/24 + 2/24 – 1/24) = 4/24 = 1/6 cistern/hour.
- Remaining work = 1 – 1/3 = 2/3 cistern.
- Time to fill remaining = (2/3) / (1/6) = 4 hours. 7. Total time = 1 (9–10 AM) + 1 (10–11 AM) + 4 = 6 hours.
- Tank full at 3 PM.

What we did and why: - Calculated work done in stages (different pipes open at different times). - Used combined rates for overlapping periods. - Added up total time to find the exact filling time.

Common Mistakes

Exam Traps

1-Minute Recap

"Listen up—this is your last-minute cheat sheet for Pipes and Cisterns. First, convert all times to rates: 1/time. Filling pipes are positive, emptying pipes are negative. Add them up for the combined rate. Time to fill = 1/combined rate. If pipes open at different times, break the problem into stages. Watch out for leaks—they’re negative rates! And if the question asks for part of the tank, multiply the total time by the fraction. That’s it. Go crush those 3–5 marks!