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There are four outlet pipes of the same capacity fixed one above the other to a water tank. The first pipe is at the bottom level and the fourth pipe is at three-fourths of the height of the tank. The third pipe is equidistant from the second pipe and the fourth pipe, the second pipe is equidistant from the first and the third. The capacity of each pipe is such that it can empty a full tank in four hours, whereas one inlet pipe can fill the empty tank in one hour. The pipe is opened to fill the empty tank and after one hour it is closed, then all the outlet pipes are opened

When the tank is full, all the four pipes are opened together and shut after one hour. If the inlet pipe is opened now, the time in which the tank will be full is ?

The solution says :

In 15 minutes, 25 liters from the tank are emptied.

In the next 20 minutes, 25 liters are emptied

In the next 30 minutes, 25 liters are emptied

Therefore, For last 25 minutes, 2 pipes are open.

In that time $\frac {25 * 25}{30}$ liters are emptied.

In an hour, $ 50 + \frac{625}{30} = 70.83 $ liters are emptied.

The inlet pipe fills the entire tank in 1 hour.

Therefore, it fills 70.83 litres in $ \frac {70.83 * 60}{100} = 42.5$ minutes

I couldn't understand why, in the last 25 minutes, $\frac {25 * 25}{30}$ liters are emptied and then why it adds it to 50 litres (though I can understand that the last two pipes will empty 50 litres in 60 minutes).

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1 Answer 1

//Edit: It's not mentioned anywhere, but you obviously set the maximum tank capacity to 100l. (For any other amount, consider the solution's results as per cent values and you'll be fine.) Just to make sure we're talking about the same things here.

The solution's logic goes like this:

1st & 2nd line combined: After 35 minutes we have 50 litres out of the tank, so another 50 litres left. At this point of time (time passed: 35 minutes), water flows out through 2 pipes, so 50 litres/hour waterflow.

3rd line: The waterflow (50 litres/hour) stays constant for the next 30 minutes, after which there would be 25 litres left, but after those 30 minutes, the total amount of time passed would be 35+30=65 minutes, pipes are closed after 60! So that 50 litres/hour waterflow only continues for 25 minutes before it is stopped.

So what we need to do now is to calculate the amount of water flowing out during those 25 minutes which are left before the pipes are closed, add it to the 50 litres which flowed out in the first 35 minutes and we get the total amount of emptied litres in one hour. We can calculate that because we made sure (3rd line) that the water is flowing out with constant speed (50 litres/hour) for the last 25 minutes. (Also for the last 30 minutes, but pipes are closed before that point of time is reached.)

So long story short: $$ \text{water leaving in last 25 minutes}=\\ =\text{25 minutes}\cdot \text{waterflow}\\=25\text{ min}\cdot 50 \frac{\mathcal{l}}{60\text{ min}}\\=25\text{ min}\cdot 25 \frac{\mathcal{l}}{30\text{ min}}\\ =25 \text{ min} \cdot \frac{25}{30}\frac{\mathcal{l}}{\text{min}}\\ =\frac{25\cdot 25}{30} \mathcal{l}$$

(As explained above, we need to add this to 50 because obviously $$\text{water leaving in 60 minutes}=\\=\text{water leaving in first 35 minutes}+\text{water leaving in last 25 minutes}=50 +\frac{25\cdot 25}{30} \mathcal{l} )$$

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