How to Solve: Pipes and Cisterns

How to Solve: Pipes and Cisterns

A Complete Guide for Students & Teachers

Introduction

"Imagine you’re filling a swimming pool for a big party—one pipe fills it in 6 hours, another in 3 hours, but a drain empties it in 12 hours. How long will it take to fill the pool? Master pipes and cisterns, and you’ll solve this—and ace your exam—in under 60 seconds."

What You Need To Know First

1. Rates of work – How much of a job (e.g., filling a tank) is done per unit time (e.g., per hour).
2. Fractions and LCM – Adding/subtracting fractions to combine rates.
3. Unitary method – Finding time taken for 1 unit of work.

Key Vocabulary

Formulas To Know

1. Rate of a Single Pipe

Formula: Rate = (1 / Time taken to fill or empty the tank)

Variables: - Rate = Fraction of the tank filled/emptied per hour. - Time = Time taken to fill/empty the tank alone (in hours).

Example: If Pipe A fills a tank in 5 hours, its rate = 1/5 tank/hour.

Memorise This.

2. Combined Rate (Multiple Pipes)

Formula: Combined Rate = Rate₁ + Rate₂ + ... ± Rateₙ (Use + for inlet pipes, – for outlet pipes.)

Variables: - Rate₁, Rate₂, ... = Rates of individual pipes.

Example: Pipe A fills at 1/4 tank/hour, Pipe B empties at 1/6 tank/hour. Combined rate = 1/4 - 1/6 = (3 - 2)/12 = 1/12 tank/hour.

Memorise This.

3. Time Taken to Fill/Empty the Tank

Formula: Time = (1 / Combined Rate)

Variables: - Time = Time taken to fill/empty the tank (in hours). - Combined Rate = Net rate of all pipes working together.

Example: If combined rate = 1/12 tank/hour, time to fill = 12 hours.

Memorise This.

4. Part of Tank Filled in Given Time

Formula: Part filled = Rate × Time

Variables: - Part filled = Fraction of the tank filled/emptied. - Rate = Rate of the pipe(s). - Time = Time pipes are open (in hours).

Example: Pipe A fills at 1/5 tank/hour. In 2 hours, it fills 1/5 × 2 = 2/5 of the tank.

Memorise This.

Step-by-Step Method

Step 1: Identify All Pipes and Their Times

• List every pipe (inlet or outlet) and the time it takes to fill or empty the tank alone.
• Example: Pipe A fills in 6 hours, Pipe B empties in 12 hours.

Step 2: Convert Times to Rates

• For each pipe, calculate its rate using Rate = 1 / Time.
• Inlet pipes = Positive rate.
• Outlet pipes = Negative rate.
• Example:
• Pipe A (inlet): 1/6 tank/hour.
• Pipe B (outlet): -1/12 tank/hour.

Step 3: Calculate Combined Rate

• Add all rates together.
• Example: 1/6 + (-1/12) = (2 - 1)/12 = 1/12 tank/hour.

Step 4: Find Time to Fill/Empty the Tank

• Use Time = 1 / Combined Rate.
• Example: 1 / (1/12) = 12 hours.

Step 5: Check for Special Conditions

• If pipes open/close at different times, break the problem into parts.
• Example: Pipe A opens for 2 hours, then Pipe B joins. Calculate separately.

Step 6: Verify Units and Answer

• Ensure time is in hours (convert minutes if needed).
• Check if the answer makes sense (e.g., combined rate should be less than the fastest pipe’s rate).

WORKED EXAMPLE (Using Steps Above)

Problem: Pipe A fills a tank in 4 hours, Pipe B fills it in 6 hours, and Pipe C empties it in 12 hours. If all three pipes are opened together, how long will it take to fill the tank?

Step 1: Identify Pipes and Times

• Pipe A (inlet): 4 hours to fill.
• Pipe B (inlet): 6 hours to fill.
• Pipe C (outlet): 12 hours to empty.

Step 2: Convert Times to Rates

• Pipe A: 1/4 tank/hour.
• Pipe B: 1/6 tank/hour.
• Pipe C: -1/12 tank/hour.

Step 3: Calculate Combined Rate

Combined Rate = 1/4 + 1/6 - 1/12 Find LCM of 4, 6, 12 = 12. = 3/12 + 2/12 - 1/12 = (3 + 2 - 1)/12 = 4/12 = 1/3 tank/hour.

Step 4: Find Time to Fill the Tank

Time = 1 / Combined Rate = 1 / (1/3) = 3 hours.

Step 5: Check for Special Conditions

• All pipes open together → No special conditions.

Step 6: Verify Answer

• Combined rate (1/3 tank/hour) is less than Pipe A’s rate (1/4 tank/hour) → Makes sense.
• Answer: 3 hours.

Worked Examples

Example 1 – Basic (Two Inlet Pipes)

Problem: Pipe A fills a tank in 3 hours, Pipe B fills it in 6 hours. How long will it take to fill the tank if both pipes are opened together?

Solution: 1. Pipe A rate = 1/3 tank/hour. 2. Pipe B rate = 1/6 tank/hour. 3. Combined rate = 1/3 + 1/6 = 1/2 tank/hour. 4. Time = 1 / (1/2) = 2 hours.

What we did and why: - Added rates because both pipes fill the tank. - Used Time = 1 / Combined Rate to find total time.

Example 2 – Medium (Inlet + Outlet Pipe)

Problem: Pipe A fills a tank in 5 hours, Pipe B empties it in 10 hours. If both are opened together, how long will it take to fill the tank?

Solution: 1. Pipe A rate = 1/5 tank/hour. 2. Pipe B rate = -1/10 tank/hour. 3. Combined rate = 1/5 - 1/10 = 1/10 tank/hour. 4. Time = 1 / (1/10) = 10 hours.

What we did and why: - Subtracted Pipe B’s rate because it empties the tank. - Combined rate is positive → Tank fills (not empties).

Example 3 – Exam Style (Pipes Open at Different Times)

Problem: A tank is filled by Pipe A in 8 hours and Pipe B in 12 hours. Pipe A is opened first, and after 2 hours, Pipe B is also opened. How long will it take to fill the tank completely?

Solution: 1. Pipe A rate = 1/8 tank/hour. 2. Pipe B rate = 1/12 tank/hour. 3. First 2 hours (only Pipe A):
- Part filled = 1/8 × 2 = 1/4 tank.
- Remaining = 1 - 1/4 = 3/4 tank. 4. After 2 hours (both pipes open):
- Combined rate = 1/8 + 1/12 = 5/24 tank/hour.
- Time to fill remaining 3/4 tank = (3/4) / (5/24) = (3/4) × (24/5) = 18/5 = 3.6 hours. 5. Total time = 2 + 3.6 = 5.6 hours (or 5 hours 36 minutes).

What we did and why: - Broke the problem into two parts (Pipe A alone, then both pipes). - Calculated remaining work after 2 hours. - Used Time = Work / Rate for the second part.

Common Mistakes

Exam Traps

1-Minute Recap

"Okay, let’s lock this in—last-minute review for pipes and cisterns. Here’s the game plan:

1. Every pipe has a rate. If it fills in 4 hours, rate = 1/4 tank/hour. If it empties in 6 hours, rate = -1/6 tank/hour.
2. Add rates for inlets, subtract for outlets. Combined rate = sum of all rates.
3. Time = 1 / Combined Rate. If combined rate is 1/3, time = 3 hours.
4. Watch for traps: Outlet pipes are negative, pipes opening late need separate steps, and always check units.
5. Practice one problem now. Grab a past paper, do it step-by-step, and you’ll own this topic.

You’ve got this—go ace that exam!