Two pipes can fill a tank in 20 and 24 minutes respectively and a waste pipe can empty 3 gallons per minute. All the three pipes working together can fill the tank in 15 minutes. The capacity of the tank is :

### Concept & Determining Waste Pipe Efficiency The capacity of a tank can be determined by finding the time it takes for a specific pipe (with a known volumetric rate) to complete its work, and then multiplying that time by the rate. $$ \text{Capacity} = \text{Time to Empty} \times \text{Volumetric Empty Rate} $$ ### Step-by-Step Solution 1. **Given**: Pipe A fills in 20 min. Pipe B fills in 24 min. Waste Pipe C empties at 3 gal/min. A + B + C fill in 15 min. 2. **Calculation**: - Let total capacity = LCM(20, 24, 15) = 120 units. - Efficiency of A = $120 / 20 = +6$ units/min. - Efficiency of B = $120 / 24 = +5$ units/min. - Net Efficiency (A + B + C) = $120 / 15 = +8$ units/min. 3. Find Efficiency of Pipe C: - $E_A + E_B + E_C = 8$ - $6 + 5 + E_C = 8$ - $11 + E_C = 8 \Rightarrow E_C = -3$ units/min. 4. Relate conceptual units to actual volume: - The waste pipe empties 3 units per minute. - We are given that it empties exactly 3 gallons per minute. - Therefore, 1 unit = 1 gallon. 5. Determine Capacity: - Total capacity is 120 units. - Since 1 unit = 1 gallon, capacity = 120 gallons. ### Exam Strategy & Shortcut Fractions approach: Waste pipe time $1/C = 1/20 + 1/24 - 1/15 = 6/120 + 5/120 - 8/120 = 3/120 = 1/40$. So it takes 40 minutes to empty. Capacity = $40 \text{ min} \times 3 \text{ gal/min} = 120 \text{ gallons}$. ### Common Pitfall A common error is confusing the net fill rate (15 minutes) as an empty rate, leading to an incorrect algebraic setup. Always note that they "can fill the tank in 15 minutes". ### Final Answer Therefore, the correct answer is **120 gallons**.