Question

Four two-way pipes A, B, C and D can either fill an empty tank or drain the full tank in 4, 10, 12 and 20 minutes respectively. All four pipes were opened simultaneously when the tank is empty. Under which of the following conditions would the tank be half filled after 30 minutes?

• A. Pipe A filled and pipes B, C and D drained
• B. Pipe A drained and pipes B, C and D filled
• C. Pipes A and D drained and pipes B and C filled
• D. Pipes A and D filled and pipes B and C drained
• E. None of the above

Hint:

For pipes-and-cistern problems, assign \(+\) for filling and \(−\) for draining. Work in a common denominator to add rates cleanly, then multiply the net rate by the time to compare with the required fraction of the tank.

Answer 1

The Correct Option is A

Step 1: Convert each pipe to a per–minute rate.
If a pipe fills (or drains) a tank in \(t\) minutes, its rate = \(\tfrac{1}{t}\) tank/min. 
A: \(\tfrac{1}{4}\), B: \(\tfrac{1}{10}\), C: \(\tfrac{1}{12}\), D: \(\tfrac{1}{20}\). 

Step 2: Check each option’s net rate and water in 30 minutes.
Take \(\mathrm{LCM}(4,10,12,20)=60\) for clean arithmetic. \[ \tfrac{1}{4}=\tfrac{15}{60},\quad \tfrac{1}{10}=\tfrac{6}{60},\quad \tfrac{1}{12}=\tfrac{5}{60},\quad \tfrac{1}{20}=\tfrac{3}{60}. \] Option A: Net rate = \(\tfrac{15}{60}-(\tfrac{6}{60}+\tfrac{5}{60}+\tfrac{3}{60})=\tfrac{15-14}{60}=\tfrac{1}{60}\) tank/min. 
In 30 minutes: \(30\times\tfrac{1}{60}=\tfrac{1}{2}\) tank \(\;\Rightarrow\;\) matches. ✔ 
Option B: Net = \((\tfrac{6+5+3}{60})-\tfrac{15}{60}=-\tfrac{1}{60}\). 
Water level decreases overall (from empty it can’t rise to \(\tfrac{1}{2}\)). ✗ 
Option C: Net = \(\tfrac{6+5}{60}-\tfrac{15+3}{60}=-\tfrac{7}{60}\). 
Negative rate \(\;\Rightarrow\;\) not possible. ✗ 
Option D: Net = \(\tfrac{15+3}{60}-\tfrac{6+5}{60}=\tfrac{7}{60}\). 
In 30 minutes: \(30\times\tfrac{7}{60}=\tfrac{7}{2}=3.5\) tanks \(\;\Rightarrow\;\) far more than half (would overflow). ✗ 

Step 3: Conclusion.
Only Option A yields exactly half a tank after 30 minutes. 

Final Answer:
\[ \boxed{\text{A. Pipe A filled; B, C, D drained}} \]