Question

The 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 third pipe working alone.

• A. If the data in statement (1) alone is sufficient to answer the question, but the data in statement (2) alone is not sufficient.
• B. If the data in statement (2) alone is sufficient to answer the question, but the data in statement (1) alone is not sufficient.
• 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.
• E. If the data in neither statement (1) nor statement (2) is sufficient to answer the question, and more data is needed.

Hint:

Set T3 = T and express T1, T2 in terms of T using each statement, then see whether the basic combined-rate equation ends up with T fully determined or not.

Answer 1

The Correct Option is B

Let the third pipe alone take T hours to fill the pool, so its rate is \(\frac{1}{T}\). The first two pipes together fill it in half that time, \(\frac{T}{2}\) hours, so their combined rate is \(\frac{2}{T}\). This gives us one basic equation: \(\frac{1}{T_1} + \frac{1}{T_2} = \frac{2}{T}\), where \(T_1\) and \(T_2\) are the times taken by the first and second pipes alone, and \(T_3 = T\).

Statement (1): \(T_1 : T_3 = 3 : 4\), so \(T_1 = \frac{3}{4}T\). Substituting into the basic equation: \(\frac{4}{3T} + \frac{1}{T_2} = \frac{2}{T}\), so \(\frac{1}{T_2} = \frac{2}{T} - \frac{4}{3T} = \frac{2}{3T}\), giving \(T_2 = \frac{3}{2}T\). Now every pipe's time is expressed as a multiple of T, but T itself is never pinned to an actual number, any value of T works (try T = 4: T1=3, T2=6, check 1/3+1/6=1/2=2/4, works; try T=8: T1=6, T2=12, check 1/6+1/12=1/4=2/8, also works). So statement (1) alone does not give a unique answer, it is not sufficient.

Statement (2): \(T_2 = T_1 + 12\) and \(T_2 = T_3 + 8 = T + 8\). From these, \(T_1 = T_2 - 12 = T - 4\). Substituting both \(T_1 = T-4\) and \(T_2 = T+8\) into the basic equation: \(\frac{1}{T-4} + \frac{1}{T+8} = \frac{2}{T}\). Combining the left side: \(\frac{(T+8)+(T-4)}{(T-4)(T+8)} = \frac{2}{T}\), i.e. \(\frac{2T+4}{T^2+4T-32} = \frac{2}{T}\). Cross multiplying: \(T(2T+4) = 2(T^2+4T-32)\), so \(2T^2+4T = 2T^2+8T-64\), giving \(4T = 64\), so \(T = 16\). Then \(T_1 = 12\) and \(T_2 = 24\). Check: \(\frac{1}{12}+\frac{1}{24} = \frac{1}{8} = \frac{2}{16}\), correct. This is now a single equation in one unknown, and it solves cleanly, so statement (2) alone is sufficient.

Statement (2) alone works, but statement (1) alone leaves T undetermined, so the correct choice is option (2).