First, let's parse the question. There are three pipes: A and C are fill pipes, and B is a drain pipe. So, A and C add water to the tank, while B removes it. The question gives me some relationships between their rates and asks for the time Pipe C takes to fill the tank alone.
Let me note down the key points:
1
Pipe B empties the tank in one hour less than Pipe A takes to fill it. So, if A takes 'a' hours, B takes 'a - 1' hours to empty.
2
When A, B, and C are all open together, the tank fills in 2 hours.
3
When only B and C are open, and after one hour B is turned off, C takes another 1 hour and 15 minutes (which is 1.25 hours) to finish filling the tank.
4
Pipe A fills the tank in less than five hours, so 'a' is less than 5.
We need to find the time Pipe C takes to fill the tank alone, which I'll denote as 'c' hours.
Alright, let's assign variables to the rates. Since rates are usually reciprocal of time, I'll let:
Rate of A = 1/a (tank per hour)
Rate of B = 1/(a - 1) (tank per hour, but since it's draining, it's negative)
Rate of C = 1/c (tank per hour)
Now, let's translate the given information into equations.
First scenario: A, B, and C together fill the tank in 2 hours. So, their combined rate is 1/2 tank per hour.
So, equation 1: (1/a) + (1/c) - (1/(a - 1)) = 1/2
Second scenario: B and C are open for one hour, then B is turned off, and C takes another 1.25 hours. So, in the first hour, B and C together fill some part, and then C fills the rest.
Compute how much is filled in the first hour. The combined rate of B and C is (1/c) - (1/(a - 1)). So, in one hour, they fill (1/c - 1/(a - 1)) tanks.
Then, the remaining part is filled by C alone in 1.25 hours. So, the remaining part is 1.25 * (1/c).
So, total filled is:
(1/c - 1/(a - 1)) + 1.25*(1/c) = 1
Let me write that as equation 2:
(1/c - 1/(a - 1)) + (5/4)*(1/c) = 1
Simplify equation 2:
Combine the terms with 1/c:
(1 + 5/4)/c - 1/(a - 1) = 1
Which is (9/4)/c - 1/(a - 1) = 1
So, equation 2 becomes: 9/(4c) - 1/(a - 1) = 1
Now, I have two equations:
1
(1/a) + (1/c) - (1/(a - 1)) = 1/2
Hmm, so I can maybe solve these two equations for a and c.
Let equation 1 as:
1/a + 1/c - 1/(a - 1) = 1/2
And equation 2 as:
9/(4c) - 1/(a - 1) = 1
Let me try to express 1/(a - 1) from equation 2.
From equation 2:
9/(4c) - 1/(a - 1) = 1
So, rearranged:
-1/(a - 1) = 1 - 9/(4c)
Multiply both sides by -1:
1/(a - 1) = -1 + 9/(4c)
So, 1/(a - 1) = (9/(4c)) - 1
Now, plug this into equation 1.
Equation 1:
1/a + 1/c - [ (9/(4c)) - 1 ] = 1/2
Simplify inside the brackets:
1/a + 1/c - 9/(4c) + 1 = 1/2
Combine like terms:
1/a + (1/c - 9/(4c)) + 1 = 1/2
Compute 1/c - 9/(4c):
(4/4c - 9/4c) = (-5)/(4c)
So, equation becomes:
1/a - 5/(4c) + 1 = 1/2
Subtract 1 from both sides:
1/a - 5/(4c) = -1/2
So, 1/a - 5/(4c) = -1/2
Let me write this as:
1/a = 5/(4c) - 1/2
Hmm, so now I have expressions for 1/(a - 1) and 1/a in terms of c.
if I can express a in terms of c or vice versa.
From equation 2, we had:
1/(a - 1) = 9/(4c) - 1
Let me write that as:
1/(a - 1) = (9 - 4c)/(4c)
So, a - 1 = 4c / (9 - 4c)
Similarly, from equation 1, we have:
1/a = 5/(4c) - 1/2
Compute 5/(4c) - 1/2:
= (5 - 2c)/(4c)
So, 1/a = (5 - 2c)/(4c)
Therefore, a = 4c / (5 - 2c)
So now, I have expressions for a and a - 1 in terms of c.
From above:
a = 4c / (5 - 2c)
And a - 1 = 4c / (9 - 4c)
So, since a - 1 = 4c / (9 - 4c), and a = 4c / (5 - 2c), we can write:
a - 1 = 4c / (9 - 4c)
But a is 4c / (5 - 2c), so:
4c / (5 - 2c) - 1 = 4c / (9 - 4c)
Let me
