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:
Pipes P, Q, R Fill Tank Time Calculation
This problem requires us to calculate the time it takes for three pipes (P, Q, and R) working together to fill an empty tank. We are given information about their individual capacities and rates, and we need to find their combined efficiency.
Step 1: Determine Pipe P's Filling Rate
Pipe P fills $\frac{2}{3}$ of the tank in 24 hours. To find its rate of filling per hour, we first determine the time it would take to fill the entire tank (1 unit capacity).
- Time for P to fill 1 tank = $24 \text{ hours} \div \frac{2}{3}$
- Time for P to fill 1 tank = $24 \times \frac{3}{2} = 36$ hours.
- So, the filling rate of Pipe P ($R_P$) is $\frac{1}{36}$ tank per hour.
Step 2: Determine Pipe Q's Filling Rate
Similarly, Pipe Q fills $\frac{2}{3}$ of the tank in 16 hours. We calculate its rate per hour.
- Time for Q to fill 1 tank = $16 \text{ hours} \div \frac{2}{3}$
- Time for Q to fill 1 tank = $16 \times \frac{3}{2} = 24$ hours.
- Thus, the filling rate of Pipe Q ($R_Q$) is $\frac{1}{24}$ tank per hour.
Step 3: Determine Pipe R's Emptying Rate
The problem provides context: P and Q were open for 4 hours, then closed, and R emptied the water in 5 hours. This information helps us find R's rate.
- First, calculate the portion of the tank filled by P and Q in 4 hours:
- Combined rate of P and Q = $R_P + R_Q = \frac{1}{36} + \frac{1}{24}$ tank/hour.
- The least common multiple (LCM) of 36 and 24 is 72.
- Combined rate of P and Q = $\frac{2}{72} + \frac{3}{72} = \frac{5}{72}$ tank/hour.
- Fraction filled in 4 hours = Rate $\times$ Time = $\frac{5}{72} \times 4 = \frac{20}{72} = \frac{5}{18}$ of the tank.
- Pipe R emptied this $\frac{5}{18}$ portion of the tank in 5 hours.
- Rate of emptying by Pipe R ($R_R$) = $\frac{\text{Amount emptied}}{\text{Time taken}}$
- $R_R = \frac{5/18 \text{ tank}}{5 \text{ hours}} = \frac{1}{18}$ tank/hour.
- Since Pipe R empties the tank, its contribution to filling is negative. So, $R_R = -\frac{1}{18}$ tank/hour.
Step 4: Calculate the Combined Rate of P, Q, and R
To find how quickly P, Q, and R together fill the tank, we sum their rates. Remember that R's rate is negative as it empties the tank.
- Combined Rate ($R_{PQR}$) = $R_P + R_Q + R_R$
- $R_{PQR} = \frac{1}{36} + \frac{1}{24} - \frac{1}{18}$ tank/hour.
- Find the LCM of the denominators (36, 24, 18), which is 72.
- $R_{PQR} = \frac{1 \times 2}{36 \times 2} + \frac{1 \times 3}{24 \times 3} - \frac{1 \times 4}{18 \times 4}$ tank/hour.
- $R_{PQR} = \frac{2}{72} + \frac{3}{72} - \frac{4}{72}$ tank/hour.
- $R_{PQR} = \frac{2 + 3 - 4}{72} = \frac{1}{72}$ tank/hour.
The resulting combined rate is positive ($\frac{1}{72}$ tank per hour), meaning the pipes working together successfully fill the tank.
Step 5: Calculate the Total Time to Fill the Tank
The time needed to fill the entire empty tank is the inverse of the combined filling rate.
- Time = $\frac{1}{\text{Combined Rate}}$
- Time = $\frac{1}{1/72 \text{ tank/hour}} = 72$ hours.
Therefore, Pipes P, Q, and R working together will take 72 hours to fill the empty tank.
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 A can fill a tank in 18 hours, pipe B can fill the same tank in 30 hours and pipe C can fill the same tank in 18 hours. The time taken by them to fill the same tank if they operate together is::
- 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?
