Question:hard

A swimming pool is fitted with three pipes. The first two pipes operating simultaneously can fill the pool in half the time taken by the third pipe alone to fill the pool. What is the time taken by the three pipes individually to fill the pool?
Statement 1: The ratio between the time taken by the first and third pipes is 3:4
Statement 2: The second pipe takes 12 hours more than the first pipe working alone and 8 hours more than the third pipe working alone

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Turn the "half the time" condition into \(\frac{1}{t_1}+\frac{1}{t_2}=\frac{2}{t_3}\), then check whether each statement fixes actual hour values or only a ratio.
Updated On: Jul 21, 2026
  • If the data in statement (1) alone is sufficient to answer the question
  • If the data in statement (2) alone is sufficient to answer the question
  • If the data in both the statements together are needed to answer the question
  • If either statement (1) alone or statement (2) alone is sufficient to answer the question
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The Correct Option is B

Solution and Explanation

Step 1: Write the pool-filling condition as a rate equation.
If pipe 1 alone takes \(t_1\) hours, pipe 2 alone takes \(t_2\) hours and pipe 3 alone takes \(t_3\) hours, their rates (fraction of pool per hour) are \(\frac{1}{t_1}\), \(\frac{1}{t_2}\) and \(\frac{1}{t_3}\). Pipes 1 and 2 together take half of \(t_3\), so their combined rate is twice pipe 3's rate: \(\frac{1}{t_1}+\frac{1}{t_2}=\frac{2}{t_3}\). This single relationship must hold no matter what the actual hours turn out to be.

Step 2: Try statement (1) on its own.
It tells us \(t_1:t_3=3:4\), a ratio and nothing else. Plugging \(t_1=3m,\ t_3=4m\) into the rate equation lets us solve \(t_2=6m\) too, but \(m\) could be 1 (giving 3, 6, 4 hours) or 2 (giving 6, 12, 8 hours), or any other positive number, and every choice still satisfies every condition given so far. Since there is no way to choose between these, statement (1) alone leaves the actual times undetermined.

Step 3: Try statement (2) on its own.
It ties pipe 2's time to the other two: \(t_2=t_1+12=t_3+8\). Substituting \(t_1=t_2-12\) and \(t_3=t_2-8\) into the rate equation and clearing denominators gives a linear equation in \(t_2\) that solves to \(t_2=24\). That immediately fixes \(t_1=12\) and \(t_3=16\) as well, three exact numbers with nothing left unknown.

Step 4: Final answer.
Only statement (2), used alone, nails down the exact time for each pipe: pipe 1 takes 12 hours, pipe 2 takes 24 hours, pipe 3 takes 16 hours. \[ \boxed{\text{Statement (2) alone is sufficient}} \]
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