Two valves $A$ and $B$ can fill an aircraft hydraulic reservoir in $10\text{ minutes}$ and $15\text{ minutes}$ respectively. If valve $A$ is opened first and the valves are opened alternately for $1\text{ minute}$ each until the tank is full, the total time taken to fill the reservoir (in minutes) is:

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### 1. Strategy & Theoretical Principles Using unit work rates with alternating cycles: - Capacity of tank = $\text{LCM}(10, 15) = 30\text{ units}$. - Rate of $A = \frac{30}{10} = 3\text{ units/min}$. - Rate of $B = \frac{30}{15} = 2\text{ units/min}$. ### 2. Analytical Formulation & Cycle Analysis In one 2-minute cycle ($A$ for 1 min, then $B$ for 1 min): $$\text{Work done in 2 minutes} = 3 + 2 = 5\text{ units}$$ ### 3. Step-by-Step Calculation Total capacity is 30 units. Number of 2-minute cycles required: $$N_{\text{cycles}} = \frac{30}{5} = 6\text{ cycles}$$ Total time elapsed: $$T = 6 \times 2 = 12\text{ minutes}$$ ### 4. Distractor Forensics - Continuous simultaneous work would take $30 / 5 = 6\text{ minutes}$. Alternating work takes exactly double ($12\text{ minutes}$).\n ### 5. Exam Trap & Standard Reference - **Trap Alert:** Always track whole cycles first before calculating remainder fractional minutes. - **Standard Reference:** Aggarwal, R.S., *Quantitative Aptitude*.

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