Pipe Filling Time: Calculating Combined Work Rate
This problem involves calculating the total time taken to fill a tank when multiple pipes work together. We need to figure out the rate at which each pipe fills the tank and then sum these rates to find the combined rate. The reciprocal of the combined rate will give us the total time taken.
Understanding Work Rate
Work rate problems are based on the idea that the amount of work done is equal to the rate of work multiplied by the time spent working (Work = Rate × Time). When multiple entities work together, their rates add up.
- If a pipe fills a tank in '$T$' hours, its rate of filling is '$\frac{1}{T}$' of the tank per hour.
Step-by-Step Calculation
-
Calculate Individual Rates:
- Pipe A fills the tank in 18 hours. Rate of Pipe A = $\frac{1}{18}$ tank/hour.
- Pipe B fills the tank in 30 hours. Rate of Pipe B = $\frac{1}{30}$ tank/hour.
- Pipe C fills the tank in 18 hours. Rate of Pipe C = $\frac{1}{18}$ tank/hour.
-
Calculate Combined Rate:
When all three pipes operate together, their rates add up. Let the combined rate be '$R_{total}$'.
$R_{total} = \text{Rate}_A + \text{Rate}_B + \text{Rate}_C$ $R_{total} = \frac{1}{18} + \frac{1}{30} + \frac{1}{18}$To add these fractions, we find the least common multiple (LCM) of the denominators (18 and 30). The LCM of 18 and 30 is 90.
$R_{total} = \frac{1 \times 5}{18 \times 5} + \frac{1 \times 3}{30 \times 3} + \frac{1 \times 5}{18 \times 5}$ $R_{total} = \frac{5}{90} + \frac{3}{90} + \frac{5}{90}$ $R_{total} = \frac{5 + 3 + 5}{90}$ $R_{total} = \frac{13}{90} \text{ tank/hour}$ -
Calculate Total Time Taken:
The time taken to fill the tank together is the reciprocal of the combined rate.
$Time_{total} = \frac{1}{R_{total}}$ $Time_{total} = \frac{1}{\frac{13}{90}}$ $Time_{total} = \frac{90}{13} \text{ hours}$ -
Convert to Mixed Fraction:
To express the time as a mixed fraction, we divide 90 by 13:
$90 \div 13 = 6 \text{ with a remainder of } 12$So, the time taken is $6\frac{12}{13}$ hours.
Final Answer Verification
The calculated time taken for pipes A, B, and C to fill the tank together is $\frac{90}{13}$ hours, which is equal to $6\frac{12}{13}$ hours. This matches one of the options provided.
Important Questions from Pipe and Cistern
Pipe P can fill $\frac{2}{3}$ part of a tank in 16 hours, and pipe Q can fill $\frac{5}{6}$ part of the same tank in 20 hours. Both P and Q were kept open for 2 hours, then both were closed. Pipe R alone was then opened and it emptied the water in the tank in 8 hours. Pipes P, Q and R together can fill the empty tank in:
Pipe P can fill $\frac{2}{3}$ part of a tank in 24 hours, and pipe Q can fill $\frac{2}{3}$ part of the same tank in 16 hours. Both P and Q were kept open for 4 hours, then both were closed. Pipe R alone was then opened and it emptied the water in the tank in 5 hours. Pipes P, Q and R together can fill the empty tank in:
- Pipe P can fill $\frac{5}{7}$ part of a tank in 30 hours, and pipe Q can fill $\frac{2}{5}$ part of the same tank in 28 hours. Both P and Q were kept open for 7 hours, then both were closed. Pipe R alone was then opened and it emptied the water in the tank in 8 hours. Pipes P, Q and R together can fill the empty tank in:
Pipe P can fill $\frac{3}{8}$ part of a tank in 21 hours and pipe Q can fill $\frac{2}{7}$ part of the same tank in 16 hours. Both P and Q were kept open for 2 hours, then both were closed. Pipe R alone was then opened and it emptied the water in the tank in 3 hours. Pipes P, Q and R together can fill the empty tank in:
- Two pipes can fill a cistern, individually, in 40 minutes and 14 minutes, respectively. There is a pipe located at the bottom of the cistern to empty it. If all the three pipes are opened simultaneously, then the empty cistern gets filled in 20 minutes. How long will the pipe at the bottom of the tank take to empty the completely filled cistern if no other pipe is then open?
