Question

Three pipes A, B and C can fill a tank in 12 hours. All the pipes started working together and after 3 hours, C is closed. If A and B can fill the remaining part in 10 hours, then the number of hours taken by C alone to fill the tank is ______.

• A. 100 hours
• B. 110 hours
• C. 120 hours
• D. 130 hours

Hint:

Find the combined rate, subtract A and B's rate found from the remaining part, to get C's rate.

Answer 1

The Correct Option is C

To solve the problem, we will go through the following steps: 

1. Determine the rate at which tanks are filled by pipes A, B, and C together.
2. Calculate the work done by pipes A, B, and C together in the initial 3 hours.
3. Identify the remaining work after C is turned off.
4. Establish the equation for the filling rates of A, B, and C using the given data.
5. Calculate the time taken by pipe C alone to fill the entire tank based on the equation.

Step 1: Rate of pipes A, B, and C together

Let the total capacity of the tank be 1 unit. Pipes A, B, and C together can fill the tank in 12 hours. Therefore, their combined rate of work is \(\frac{1}{12}\) of the tank per hour.

Step 2: Work done by A, B, and C in 3 hours

In 3 hours, together they would fill: 
\(3 \times \frac{1}{12} = \frac{1}{4}\) of the tank.

Step 3: Remaining work

The remaining part of the tank to be filled after 3 hours is:

\(1 - \frac{1}{4} = \frac{3}{4}\) of the tank.

Step 4: Work done by A and B in 10 hours

Given that A and B can fill the remaining \(\frac{3}{4}\) of the tank in 10 hours:

The rate of A and B combined is:

\(\frac{3}{4} \div 10 = \frac{3}{40}\) of the tank per hour.

Step 5: Establish the equation and solving for C

The rate of A, B, and C together is \(\frac{1}{12}\), thus:

\(\frac{1}{12} = \frac{3}{40} + \text{Rate of C}\)

Simplifying the equation for the rate of C:

\(\text{Rate of C} = \frac{1}{12} - \frac{3}{40}\)

Convert these into equivalent fractions with a common denominator:

• \(\frac{1}{12} = \frac{10}{120}\)
• \(\frac{3}{40} = \frac{9}{120}\)

Thus,

\(\text{Rate of C} = \frac{10}{120} - \frac{9}{120} = \frac{1}{120}\)

This means pipe C alone can fill the tank in 120 hours.

Therefore, the correct answer is 120 hours.

Answer 2

Step 1: Write the combined rate of all three pipes.
Taking the tank as 1 full job, A, B and C together fill \(1/12\) of the tank every hour, since they finish it in 12 hours working together.

Step 2: Find how much is done in the first 3 hours.
Working together for 3 hours fills \(3 \times \dfrac{1}{12} = \dfrac{1}{4}\) of the tank. So \(\dfrac{3}{4}\) of the tank is still empty when C is closed.

Step 3: Use the A and B rate to check the remaining part.
A and B alone fill this remaining \(\dfrac{3}{4}\) in 10 hours, so their combined rate is:
\[ a+b = \frac{3/4}{10} = \frac{3}{40} \]

Step 4: Subtract to get C's rate alone.
\(c = (a+b+c) - (a+b) = \dfrac{1}{12} - \dfrac{3}{40}\). Using 120 as the common denominator: \(\dfrac{10}{120} - \dfrac{9}{120} = \dfrac{1}{120}\).

Step 5: Convert C's rate into time.
If C fills \(1/120\) of the tank every hour, C alone takes 120 hours to fill it fully.

Final Answer:
C alone takes 120 hours to fill the tank.
\[ \boxed{120 \text{ hours}} \]