A tank has four pipes $P_1$, $P_2$, $P_3$ and $P_4$. The tank can be filled in $15$ minutes by pipes $P_1$, $P_2$, $P_3$ together. It can be filled in $20$ minutes by pipes $P_2$, $P_3$, $P_4$ together and it can be filled by pipes $P_1$, $P_4$ together in $30$ minutes. If all the pipes are opened together, then in how much time will the tank be filled?
Understanding Pipe Filling Rates
This problem involves calculating the time it takes to fill a tank when different combinations of pipes are used. We need to find the combined rate of all four pipes ($P_1, P_2, P_3, P_4$) when working together.
Setting Up Equations for Pipe Rates
Let the rates at which pipes $P_1, P_2, P_3,$ and $P_4$ fill the tank be $r_1, r_2, r_3,$ and $r_4$ respectively. The rate is measured in terms of the fraction of the tank filled per minute.
- Pipes $P_1, P_2, P_3$ together fill the tank in $15$ minutes. Their combined rate is: $r_1 + r_2 + r_3 = \frac{1}{15} \quad (\text{tank/min})$
- Pipes $P_2, P_3, P_4$ together fill the tank in $20$ minutes. Their combined rate is: $r_2 + r_3 + r_4 = \frac{1}{20} \quad (\text{tank/min})$
- Pipes $P_1, P_4$ together fill the tank in $30$ minutes. Their combined rate is: $r_1 + r_4 = \frac{1}{30} \quad (\text{tank/min})$
Calculating the Combined Rate of All Pipes
We want to find the time taken when all pipes ($P_1, P_2, P_3, P_4$) are opened together. Let the time taken be $T$ minutes. The combined rate of all pipes is $r_1 + r_2 + r_3 + r_4 = \frac{1}{T}$.
To find this combined rate, let's add the three given equations:
$ (r_1 + r_2 + r_3) + (r_2 + r_3 + r_4) + (r_1 + r_4) = \frac{1}{15} + \frac{1}{20} + \frac{1}{30} $Combine the terms on the left side:
$ 2r_1 + 2r_2 + 2r_3 + 2r_4 = 2(r_1 + r_2 + r_3 + r_4) $Find a common denominator for the fractions on the right side (the least common multiple of $15, 20,$ and $30$ is $60$):
$ \frac{1}{15} + \frac{1}{20} + \frac{1}{30} = \frac{4}{60} + \frac{3}{60} + \frac{2}{60} = \frac{4+3+2}{60} = \frac{9}{60} $Simplify the fraction:
$ \frac{9}{60} = \frac{3}{20} $Now, equate the combined rates:
$ 2(r_1 + r_2 + r_3 + r_4) = \frac{3}{20} $Solve for the combined rate of all four pipes:
$ r_1 + r_2 + r_3 + r_4 = \frac{3}{20 \times 2} = \frac{3}{40} \quad (\text{tank/min}) $Determining the Total Time to Fill the Tank
The time $T$ taken to fill the tank when all pipes are open together is the reciprocal of their combined rate:
$ T = \frac{1}{\text{Combined Rate}} = \frac{1}{3/40} = \frac{40}{3} \text{ minutes} $To express this time in minutes and seconds:
$ \frac{40}{3} \text{ minutes} = 13 \frac{1}{3} \text{ minutes} $Convert the fractional part of the minute to seconds:
$ \frac{1}{3} \text{ minute} = \frac{1}{3} \times 60 \text{ seconds} = 20 \text{ seconds} $Therefore, the tank will be filled in $13$ minutes and $20$ seconds when all pipes are opened together.
Important Questions from Time & Work (Notes)
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- $6$ men or $5$ women earn ₹ $16,800$ in $4$ days. How much will $4$ women and $6$ men earn in one day?
