To solve the problem step by step, we will follow the logic laid out in the video transcript.
### Step 1: Determine the total work done by all pipes together.
Given that pipes A, B, and C together can fill a cistern in 12 hours, we can define the total work as 12 units (where 1 unit represents the entire cistern).
**Hint:** The total work done can be thought of as the total volume of the cistern, which is filled in a specific time.
### Step 2: Calculate the work done by A, B, and C in 4 hours.
If they can fill the cistern in 12 hours, the work done by A, B, and C in one hour is:
\[
\text{Work done in 1 hour} = \frac{1 \text{ cistern}}{12 \text{ hours}} = \frac{1}{12} \text{ cistern/hour}
\]
In 4 hours, the work done will be:
\[
\text{Work done in 4 hours} = 4 \times \frac{1}{12} = \frac{4}{12} = \frac{1}{3} \text{ cistern}
\]
**Hint:** Multiply the hourly work rate by the number of hours to find the total work done in that time.
### Step 3: Calculate the remaining work after 4 hours.
The remaining work after 4 hours is:
\[
\text{Remaining work} = 1 - \frac{1}{3} = \frac{2}{3} \text{ cistern}
\]
**Hint:** Subtract the work done from the total work to find out how much is left.
### Step 4: Determine the efficiency of A and B together.
A and B together take 10 hours to fill the remaining \(\frac{2}{3}\) of the cistern. Therefore, their combined efficiency can be calculated as follows:
\[
\text{Efficiency of A + B} = \frac{\text{Remaining work}}{\text{Time taken}} = \frac{\frac{2}{3}}{10} = \frac{2}{30} = \frac{1}{15} \text{ cistern/hour}
\]
**Hint:** Efficiency is calculated by dividing the amount of work done by the time taken.
### Step 5: Calculate the efficiency of C.
The combined efficiency of A, B, and C is:
\[
\text{Efficiency of A + B + C} = \frac{1}{12} \text{ cistern/hour}
\]
Now, we can find the efficiency of C:
\[
\text{Efficiency of C} = \text{Efficiency of A + B + C} - \text{Efficiency of A + B} = \frac{1}{12} - \frac{1}{15}
\]
To perform this subtraction, we need a common denominator (which is 60):
\[
\frac{1}{12} = \frac{5}{60}, \quad \frac{1}{15} = \frac{4}{60}
\]
Thus,
\[
\text{Efficiency of C} = \frac{5}{60} - \frac{4}{60} = \frac{1}{60} \text{ cistern/hour}
\]
**Hint:** Use a common denominator to subtract fractions.
### Step 6: Calculate how long C will take to fill two-thirds of the cistern.
To find out how long C will take to fill \(\frac{2}{3}\) of the cistern, we use the formula:
\[
\text{Time} = \frac{\text{Work}}{\text{Efficiency}} = \frac{\frac{2}{3}}{\frac{1}{60}} = \frac{2}{3} \times 60 = 40 \text{ hours}
\]
**Hint:** Time taken can be calculated by dividing the amount of work by the efficiency.
### Final Answer:
C alone will fill two-thirds of the cistern in **40 hours**.
