Calculating Combined Pipe Filling Time
This problem involves finding the time it takes for multiple pipes operating simultaneously to fill a tank, given their individual filling times.
Individual Pipe Rates
First, determine the rate at which each pipe fills the tank. The rate is the fraction of the tank filled per hour.
- Rate of Pipe A: $\frac{1}{19}$ tank per hour.
- Rate of Pipe B: $\frac{1}{21}$ tank per hour.
- Rate of Pipe C: $\frac{1}{8}$ tank per hour.
Combined Rate Calculation
To find the combined rate when all pipes operate together, add their individual rates.
Combined Rate = Rate A + Rate B + Rate C
Combined Rate = $\frac{1}{19} + \frac{1}{21} + \frac{1}{8}$
Find a common denominator, which is the least common multiple (LCM) of 19, 21, and 8. LCM(19, 21, 8) = $19 \times 21 \times 8 = 3192$.
Combined Rate = $\frac{1 \times (21 \times 8)}{19 \times 21 \times 8} + \frac{1 \times (19 \times 8)}{19 \times 21 \times 8} + \frac{1 \times (19 \times 21)}{19 \times 21 \times 8}$
Combined Rate = $\frac{168}{3192} + \frac{152}{3192} + \frac{399}{3192}$
Combined Rate = $\frac{168 + 152 + 399}{3192} = \frac{719}{3192}$ tank per hour.
Total Time Calculation
The time taken to fill the tank together is the reciprocal of the combined rate.
Time = $\frac{1}{\text{Combined Rate}}$
Time = $\frac{1}{\frac{719}{3192}} = \frac{3192}{719}$ hours.
Convert to Mixed Fraction
Convert the improper fraction $\frac{3192}{719}$ into a mixed number.
Divide 3192 by 719:
$3192 \div 719 = 4$ with a remainder of $3192 - (4 \times 719) = 3192 - 2876 = 316$.
Therefore, the time taken is $4 \frac{316}{719}$ hours.
Similar Questions
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Important Questions from Pipe and Cistern
Pipes A, B and C can fill a tank in 20, 30 and 60 hours, respectively. Pipes A, B and C are opened at 7 a.m., 8 a.m., and 9 a.m., respectively, on the same day. When will the tank be full?
There are two water taps in a tank which can fill the empty tank in 12 hours and 18 hours respectively. It is seen that there is a leakage point at the bottom of the tank which can empty the completely filled tank in 36 hours. If both the water taps are opened at the same time to fill the empty tank and leakage point was repaired after 1 hour, then in how much time the empty tank will be completely filled?
Two pipes A and B can fill a tank in 12 minutes and 24 minutes, respectively, while a third pipe C can empty the full tank in 32 minutes. All the three pipes are opened simultaneously. However, pipe C is closed 2 minutes before the tank is filled. In how much time (in minutes) will the tank be full?
Pipes A and B can fill a tank in 12 hours and 16 hours respectively and pipe C can empty the full tank in 24 hours. All three pipes are opened together, but after 4 hours pipe B is closed. In how many hours, the empty tank will be completely filled?
Pipes A and B can fill a tank in 43.2 minutes and 108 minutes, respectively. Pipe C can empty it at 3 litres/minute. When all the three pipes are opened together, they fill the tank in 54 minutes. The capacity (in litres) of the tank is:
