Introduction / Context:
This problem involves three pipes with different filling speeds. Their relative speeds are expressed using ratios. The tank filling time when all pipes work together is given, and we are asked to find the time taken by the slowest pipe working alone. Such questions are very common in competitive exams and test understanding of work rate relations and proportional reasoning between different agents working together.
Given Data / Assumptions:
Three inlet pipes X, Y and Z together fill the tank in 5 hours.
Pipe Z is three times as fast as pipe Y.
Pipe Y is twice as fast as pipe X.
All pipes are assumed to have constant flow rates.
Concept / Approach:The best approach is to express all rates in terms of the slowest pipe, say pipe X. From the relative speed information, we write Y and Z in multiples of X. The combined rate is the sum of these three rates and is equal to one tank divided by the total time when all three work together. This allows us to solve for the rate of X and then find the time taken by pipe X alone to fill the tank. Using consistent units (hours) throughout prevents confusion.
Step-by-Step Solution:Let the rate of pipe X be r tanks per hour.Given that Y is twice as fast as X, the rate of Y is 2r tanks per hour.Given that Z is three times as fast as Y, the rate of Z is 3 * 2r = 6r tanks per hour.Therefore, the combined rate of X, Y and Z is r + 2r + 6r = 9r tanks per hour.Together they fill the tank in 5 hours, so 9r = 1/5 tank per hour.Thus r = 1 / (5 * 9) = 1/45 tank per hour.The time taken by pipe X alone to fill the tank is 1 / r = 45 hours.Verification / Alternative check:Check how many tanks the three pipes fill in 5 hours using r = 1/45.Rate of X = 1/45, rate of Y = 2/45, rate of Z = 6/45.Combined rate = (1 + 2 + 6) / 45 = 9/45 = 1/5 tank per hour.In 5 hours, they fill (1/5) * 5 = 1 tank, which matches the question. So the calculation is consistent.Why Other Options Are Wrong:30 hours and 40 hours are too small and imply a faster pipe X, which would make the total filling time with all three pipes less than 5 hours. On the other hand, 60 hours and 75 hours are too large and would slow down pipe X too much, meaning the three pipes together would take more than 5 hours to fill the tank. Only 45 hours fits the given combined time and the specified speed ratios.
Common Pitfalls:Students often misread the relative speed relations, for example treating “Z is three times as fast as Y” as Z = 3 times X. Another common mistake is to assume that time, rather than rate, is directly proportional, leading to incorrect equations like Tz = 3 * Ty. The correct approach is always to express rates, not times, in the stated ratios. Careful translation of the word relations into algebraic expressions avoids these issues and ensures that work rates add correctly.
Final Answer:Pipe X alone will fill the water tank in 45 hours.
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