Question

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

• A. If the data in statement (1) alone is sufficient to answer the question
• B. If the data in statement (2) alone is sufficient to answer the question
• C. If the data in both the statements together are needed to answer the question
• D. If either statement (1) alone or statement (2) alone is sufficient to answer the question

Hint:

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.

Answer 1

The Correct Option is B

Step 1: Translate the main condition into an equation.
Let the first, second and third pipes alone take \(t_1\), \(t_2\) and \(t_3\) hours respectively to fill the pool. The first two pipes together fill the pool in half the time the third pipe alone takes, so their combined time is \(\frac{t_3}{2}\), meaning their combined rate is \(\frac{2}{t_3}\). In terms of individual rates, \[ \frac{1}{t_1}+\frac{1}{t_2}=\frac{2}{t_3} \] This one equation connects all three unknowns, and enough extra information is needed to solve for \(t_1\), \(t_2\) and \(t_3\) as actual numbers.

Step 2: Check statement (1) alone.
Statement (1) only gives a ratio, \(t_1:t_3=3:4\), so write \(t_1=3m\) and \(t_3=4m\) for an unknown scale \(m\). Substituting into the main equation gives \(\frac{1}{3m}+\frac{1}{t_2}=\frac{2}{4m}=\frac{1}{2m}\), so \(\frac{1}{t_2}=\frac{1}{2m}-\frac{1}{3m}=\frac{1}{6m}\), meaning \(t_2=6m\). All three times come out as multiples of \(m\) (\(3m\), \(6m\), \(4m\)), but \(m\) itself is never fixed to an actual number of hours. Statement (1) alone cannot give the actual times.

Step 3: Check statement (2) alone.
Statement (2) gives \(t_2=t_1+12\) and \(t_2=t_3+8\), so \(t_1=t_2-12\) and \(t_3=t_2-8\). Substitute these into the main equation, writing everything in terms of \(t_2\) (call it \(x\)): \[ \frac{1}{x-12}+\frac{1}{x}=\frac{2}{x-8} \] Multiplying through by \(x(x-12)(x-8)\) and simplifying leads to \(96=4x\), so \(x=24\). Then \(t_1=24-12=12\) hours, \(t_2=24\) hours and \(t_3=24-8=16\) hours. Every one of the three times comes out as an exact number of hours using statement (2) alone.

Step 4: Final answer.
Statement (1) alone only fixes the ratio of the times, not their actual values, so it is not sufficient by itself. Statement (2) alone solves completely for \(t_1=12\), \(t_2=24\) and \(t_3=16\) hours. \[ \boxed{\text{Statement (2) alone is sufficient}} \]