A large fresh water reservoir is fitted with two types of feeder pipes – hot water pipes and cold water pipes. Six cold water pipes alone can fill the reservoir in 12 hours. 3 cold water pipes and 9 hot water pipes together can fill the reservoir in 8 hours. How long will 5 hot water pipes alone take to fill the reservoir?

### Concept & Logic To find the time taken by the hot water pipes, we need to determine the filling rate (work done per hour) for both a single hot water pipe and a single cold water pipe. We can set up a system of linear equations based on their combined rates. ### Step-by-Step Solution 1. Let the rate of 1 cold water pipe be $C$ and the rate of 1 hot water pipe be $H$. 2. 6 cold water pipes can fill the reservoir in 12 hours. $$6C = \frac{1}{12}$$ $$C = \frac{1}{72} \text{ (Reservoir per hour)}$$ 3. 3 cold water pipes and 9 hot water pipes can fill the reservoir in 8 hours. $$3C + 9H = \frac{1}{8}$$ 4. Substitute the value of $C$ into the second equation: $$3 \left(\frac{1}{72}\right) + 9H = \frac{1}{8}$$ $$\frac{1}{24} + 9H = \frac{1}{8}$$ 5. Solve for $H$: $$9H = \frac{1}{8} - \frac{1}{24}$$ $$9H = \frac{3 - 1}{24} = \frac{2}{24} = \frac{1}{12}$$ $$H = \frac{1}{108} \text{ (Reservoir per hour)}$$ 6. Now, find the rate of 5 hot water pipes: $$5H = 5 \times \frac{1}{108} = \frac{5}{108}$$ 7. The time taken by 5 hot water pipes alone is the reciprocal of their combined rate: $$\text{Time} = \frac{108}{5} \text{ hours} = 21.6 \text{ hours}$$ 8. Convert the decimal part into minutes: $$0.6 \text{ hours} = 0.6 \times 60 \text{ minutes} = 36 \text{ minutes}$$ Total time = 21 hrs 36 min. ### Exam Strategy & Shortcut Instead of finding individual rates immediately, you can substitute the group of cold pipes directly. Since 6 cold pipes take 12 hours, 3 cold pipes take 24 hours. Their rate is $1/24$. Subtracting this from the combined rate ($1/8$) gives the rate of 9 hot pipes. ### Common Pitfall Students often forget to convert the decimal fraction of an hour ($0.6$) into minutes correctly, sometimes mistakenly writing it as 60 minutes or leaving it as a decimal. ### Final Answer Therefore, the correct answer is **21 hrs 36 min**.