A tank fitted with two pipes is to be filled with water. The first pipe can fill it in 10 hours. After it has been open for 5 hours, the second pipe is opened and the tank is filled in 2 hours more. How long will it take the second pipe to fill the tank alone?

Let's break down the information given step-by-step:

Step 1: The first pipe can fill the tank in 10 hours.

Step 2: After the first pipe has been open for 5 hours, the second pipe is opened.

Step 3: The tank is then filled in 2 hours after the second pipe is opened.

Now, let's calculate how much of the tank is filled by the first pipe in 1 hour.

If the first pipe can fill the tank in 10 hours, then in 1 hour, it fills 1/10 of the tank.

Since the first pipe is open for 5 hours, it fills (1/10) × 5 = 1/2 of the tank.

Next, let's find out how much of the tank is filled by the second pipe in 1 hour.

Since the first pipe has already filled 1/2 of the tank, the remaining 1/2 of the tank is filled in the remaining time, which is 2 hours.

Therefore, the second pipe fills 1/2 ÷ 2 = 1/4 of the tank in 1 hour.

Now, to determine how long the second pipe would take to fill the tank alone, let's calculate it:

If the second pipe fills 1/4 of the tank in 1 hour, it would take 4 hours to fill the whole tank by itself.

So, the second pipe would take 4 hours to fill the tank alone.

To solve this problem, we can break it down into smaller parts and use the concept of work done.

Let's assume that the tank has a total capacity of 1 unit.

In the first 5 hours, the first pipe fills 1/10th of the tank per hour.
Therefore, in 5 hours, the first pipe fills 5/10th or 1/2 of the tank.

After 5 hours, the tank is already half full. So, the remaining half of the tank needs to be filled in 2 additional hours.

Let's denote the rate at which the second pipe fills the tank as 'x' (in units per hour).

In these 2 hours, the combined rate of both pipes is 1/2 of the tank.
So, we can write the equation as:

5/10 + (2 * x) = 1

Simplifying the equation:

1/2 + 2x = 1
2x = 1 - 1/2
2x = 1/2
x = 1/4

From the equation, we find that 'x' (the rate of the second pipe) is 1/4 or 0.25 units per hour.

Since the second pipe fills the entire tank alone, we can answer the question.

Therefore, it would take the second pipe 1 / 0.25 = 4 hours to fill the tank alone.

(1/10)(5)+(2)[(1/10)+(1/x)] = 1

x = 20/3 hrs

After 5 hours, the tank is half full, so

1/10 + 1/x = (1/2)/2
x = 20/3 hr

check:
pipe 1 can fill 1/10 of the tank per hour
pipe 2 can fill 3/20 of the tank per hour
5/10 + 2(1/10 + 3/20) = 5/10 + 10/20 = 1