Pipe A can fill an empty tank in 18 hours and pipe B can fill the same empty tank in 24 hours. If both the pipes are opened simultaneously, how much time (in hours) will they take to fill the empty tank?
Calculating Combined Pipe Filling Time
This problem involves calculating the time it takes for two pipes, A and B, working together to fill an empty tank. We are given the individual times each pipe takes to fill the tank.
Understanding Pipe Filling Rates
To solve this, we first determine the rate at which each pipe fills the tank. The rate is the fraction of the tank filled per hour.
- Pipe A fills the tank in 18 hours. So, the rate of Pipe A is $\frac{1}{18}$ of the tank per hour.
- Pipe B fills the tank in 24 hours. So, the rate of Pipe B is $\frac{1}{24}$ of the tank per hour.
Calculating Combined Filling Rate
When both pipes are opened simultaneously, their rates add up. We need to find the combined rate:
Combined Rate = Rate of Pipe A + Rate of Pipe B
Combined Rate = $ \frac{1}{18} + \frac{1}{24} $
To add these fractions, we find a common denominator. The least common multiple (LCM) of 18 and 24 is 72.
Combined Rate = $ \frac{1 \times 4}{18 \times 4} + \frac{1 \times 3}{24 \times 3} $
Combined Rate = $ \frac{4}{72} + \frac{3}{72} $
Combined Rate = $ \frac{4 + 3}{72} $
Combined Rate = $ \frac{7}{72} $ of the tank per hour.
Determining Total Time Taken Together
The time taken to fill the tank together is the reciprocal of the combined rate.
Time = $ \frac{1}{\text{Combined Rate}} $
Time = $ \frac{1}{\frac{7}{72}} $
Time = $ \frac{72}{7} $ hours.
Converting Time to Mixed Fraction
To express the answer in the format given in the options, we convert the improper fraction $ \frac{72}{7} $ to a mixed number:
Divide 72 by 7:
$ 72 \div 7 = 10 $ with a remainder of $ 2 $.
So, $ \frac{72}{7} $ hours is equal to $ 10\frac{2}{7} $ hours.
Final Answer Analysis
The time calculated, $ 10\frac{2}{7} $ hours, matches one of the options provided.
Similar Questions
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Important Questions from Pipe and Cistern
A pipe can fill a cistern in 20 minutes where as the cistern when full can be emptied by a leak in 28 minutes. When both are opened, The time taken to fill the cistern is:
‘A’ pipe can empty a tank in 20 minutes. The second pipe ‘B’ has a diameter twice as that of ‘A’. If both A & B pipe are attached to the tank how much time will be required to empty the tank?
Pipes A and B can empty a full tank in 16 hours and 24 hours, respectively. Pipe C alone can fill the empty tank in 4 hours. If A, B and C are opened together, the tank will be 35% full after :
Pipes A and B can fill a tank in 36 minutes and 45 minutes, respectively. Both these pipes were opened simultaneously. After 20 minutes, a leak at the bottom of the tank was spotted which was immediately sealed. The tank was full in another 15 minutes. The leak alone can empty the full tank in:
A cistern has a leak which would empty it in 6 hours. A tap is turned on which admits 10 litres of water per minute into the cistern. When it is full it is now emptied in 10 hours. What is the capacity (in litres) of the cistern?
