Leak in Cistern Problems with Filling Time Calculations

Leak in Cistern problems involve a tank being filled by one or more pipes while an outlet or leak removes water. Solve them by expressing each pipe as a fraction of the tank filled or emptied in one hour, then finding the net rate and total time.

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What Is a Leak in Cistern?

A leak in a cistern removes water while an inlet pipe fills the tank. The effective rate is the filling rate minus the leakage rate.

If an inlet fills a tank in x hours and a leak empties it in y hours, the inlet rate is 1/x tank per hour and the leakage rate is 1/y tank per hour. Therefore, the net rate is 1/x − 1/y. The tank fills only when the inlet rate is greater than the leakage rate.

Leak in Cistern Video Lessons

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Pipes and Leaks: Tank Filling Fractions

Understand how leaks affect pipe efficiency and calculate the fraction of a tank filled or emptied using time-and-work methods.

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Leak in Cistern Formula & Tricks

Important Formulas

Net filling rate
Net rate = 1/x − 1/y

Here, x is the time taken by the inlet alone and y is the time taken by the leak to empty the full cistern.

Time to fill with a leak
T = 1 ÷ (1/x − 1/y) = xy/(y − x)

This formula applies when y > x, so the inlet is faster than the leak.

Time taken by the leak alone
1/y = 1/x − 1/T

If the inlet fills the tank in x hours and the inlet with the leak fills it in T hours, this gives the leak's emptying rate.

Leak time from combined time
y = xT/(T − x)

This is obtained from 1/y = 1/x − 1/T. The combined time T must be greater than x.

Quick Tricks

Use the LCM as the tank capacity

When the inlet and leak times are x and y hours, take the tank capacity as LCM(x, y) units. Convert each rate into units per hour and subtract.

Example: For x = 6 and y = 9, take capacity 18 units. Inlet rate = 3 units/hour, leak rate = 2 units/hour, so net rate = 1 unit/hour and filling time = 18 hours.
Check the direction of the time change

A leak must make the filling time greater than the inlet-only time. If the calculated combined time is smaller, the sign has been used incorrectly.

Example: An inlet fills a tank in 8 hours. With a leak, the time cannot be less than 8 hours.
Compare rates before calculating

If the leak empties the cistern in less time than the inlet takes to fill it, the leak is stronger and the tank cannot fill through that inlet alone.

Example: An inlet fills in 10 hours and a leak empties in 6 hours. Since 1/6 > 1/10, the net rate is negative and the tank does not fill.

Leak in Cistern Concepts

Converting Filling and Leakage Time into Rates

A pipe that fills a cistern in x hours fills 1/x of the tank in one hour, while a leak that empties it in y hours removes 1/y of the tank per hour.

The complete cistern is treated as one unit. For example, if a pipe fills the tank in 12 hours, its rate is 1/12 tank per hour. If a leak empties it in 18 hours, its rate is 1/18 tank per hour.

Example: A pipe fills 1/8 of a tank per hour and a leak removes 1/12 per hour. Their combined rate is 1/8 − 1/12 = 1/24 tank per hour.

Finding the Filling Time with a Leak

Subtract the leak rate from the inlet rate and take the reciprocal of the remaining rate.

If the inlet fills in x hours and the leak empties in y hours, the net rate is (y − x)/(xy). Hence, the time is xy/(y − x), provided y > x.

Example: An inlet fills a tank in 6 hours and a leak empties it in 9 hours. Net rate = 1/6 − 1/9 = 1/18, so the tank fills in 18 hours.

Finding the Leak Rate from a Delayed Filling Time

If the inlet alone fills the tank in x hours and the inlet with the leak takes T hours, the leak rate is 1/x − 1/T.

The leak causes the difference between the inlet-only rate and the combined rate. The leak's emptying time is y = xT/(T − x). Since the leak slows filling, T must be greater than x.

Example: A pipe fills a tank in 10 hours, but takes 15 hours with a leak. Leak rate = 1/10 − 1/15 = 1/30, so the leak alone empties the tank in 30 hours.

When the Tank Cannot Fill

The cistern cannot fill if the leakage rate is equal to or greater than the inlet rate.

If x is the inlet time and y is the leakage time, then x ≥ y means 1/x ≤ 1/y. For x = y, the water level remains unchanged; for x > y, the tank loses water whenever the inlet is open.

Example: An inlet fills a tank in 8 hours and a leak empties it in 8 hours. Net rate = 1/8 − 1/8 = 0, so the tank never fills.

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Practice Leak in Cistern Questions

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1The price of petrol is increased by 28%. A person wants to increase his expenditure by 22% only. By approximately what percent should he decrease his consumption?→ 2An article is sold for Rs. 288 after successive discounts of 20% and 25%. What is the marked price of the article?→ 321.6 ÷ 3.6 × 2 + 0.25 × 16 ÷ 4 - 6 is equal to∶→ 4What is the value of x so that the seven digit number 55350x2 is divisible by 72?→ 5Two articles are sold for Rs. 4,752 each. On one, the seller gains 32% and on the other he loses 28%. What is his overall gain or loss percentage, correct to one decimal place?→ 6The price of sugar is increased by 17%. A person wants to increase his expenditure by 5% only. By approximately what percent should he decrease his consumption?→ 7The value of 15.2 + 5.8 ÷ 2.9 × 2 - 3.5 × 2 ÷ 0.5 is equal to:→ 8What is the value of x so that the seven digit number 8439x53 is divisible by 99?→ 9Two articles are sold for Rs. 962 each. On one, the seller gains 30% and on the other he loses 26%. What is his overall gain or loss percentage, nearest to one decimal place?→ 10An article is sold for Rs. 288 after successive discounts of 25% and x%. If the marked price of the article is Rs. 480, what is the value of x?→

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Quick Revision Notes

Leak in Cistern Revision Points

Use rates of tank capacity per hour and subtract the leakage rate from the filling rate.

  • Inlet rate = 1/(time taken by inlet).
  • Leakage rate = 1/(time taken by leak).
  • Net rate = inlet rate − leakage rate.
  • Time = 1/net rate when the net rate is positive.
  • For inlet time x and leak time y, combined time = xy/(y − x).
  • The combined filling time must be greater than the inlet-only time.
  • If the leak rate is equal to or greater than the inlet rate, the tank does not fill.

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Leak in Cistern FAQs

What is the leak in cistern formula?

If an inlet fills a tank in x hours and a leak empties it in y hours, the filling time is xy/(y − x), where y > x.

A pipe fills a cistern in 5 hours and a leak empties it in 20 hours. How long will the cistern take to fill?

Net rate = 1/5 − 1/20 = 3/20 tank per hour. Time = 20/3 hours, or 6 hours 40 minutes.

A pipe fills a tank in 12 hours, and with a leak it fills in 18 hours. In how much time will the leak empty the full tank?

Leak rate = 1/12 − 1/18 = 1/36. Therefore, the leak alone empties the tank in 36 hours.

Can a tank fill if the inlet takes 7 hours and the leak takes 5 hours to empty it?

No. The leak rate, 1/5, is greater than the inlet rate, 1/7. The net rate is negative.

What happens when the inlet and leak have equal rates?

The net rate is zero. The water level remains constant while both the inlet and leak are open, so the tank does not fill.

Why is the filling time with a leak greater than the inlet-only time?

The leak reduces the net rate. Since time is the reciprocal of the positive net rate, the reduced rate produces a longer filling time.

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