Pipes & Cisterns

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Quick recall

• Add filling rates; subtract emptying rates.
• A tank fills only when the net filling rate is positive.

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Formula Cards

1. Net filling rate

$$R=R_{\text{fill}}-R_{\text{empty}}$$

Add all filling rates and subtract all emptying rates.

2. Time to fill an empty tank

$$T=\frac{1}{R}$$

Requires R > 0; rates are tank fractions per hour.

3. One pipe and one leak

$$T=\frac{xy}{x-y}$$

Leak empties a full tank in x hours; pipe fills it in y hours. Requires y < x.

Worked Examples

Example 1

Question: A pipe fills a tank in 6 hours; a leak empties it in 12 hours. Find net filling time.

Step 1: Subtract the emptying rate.

$$R=\frac{1}{6}-\frac{1}{12}=\frac{1}{12}$$

Step 2: Invert the positive net rate.

$$T=12\text{ hours}$$

Answer: 12 hours.

Example 2

Question: Pipes fill in 10 and 15 hours; a leak empties in 30 hours. Find filling time.

Step 1: Add inflows and subtract outflow.

$$R=\frac{1}{10}+\frac{1}{15}-\frac{1}{30}=\frac{2}{15}$$

Step 2: Invert.

$$T=\frac{15}{2}=7.5\text{ hours}$$

Answer: 7.5 hours.

Example 3

Question: A pipe fills in 4 hours, but with a leak takes 5 hours. Find the leak's emptying time.

Step 1: Find the lost rate.

$$R_{\text{leak}}=\frac{1}{4}-\frac{1}{5}=\frac{1}{20}$$

Step 2: Invert the leak rate.

$$T_{\text{leak}}=20\text{ hours}$$

Answer: 20 hours.

Drill

Try each question before opening its answer.

Question 1

Pipe fills in 5h, leak empties in 10h. Net fill time?

• Show answer

  10h

Question 2

Two pipes fill in 12h and 18h together. Time?

• Show answer

  7.2h

Question 3

A pipe fills a tank in 8h; with a leak it takes 10h. Leak alone empties in?

• Show answer

  40h

Question 4

Three pipes fill in 6h, 8h, 12h. Together?

• Show answer

  2.67h

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