A tap fills a tank in 15 hours. Another tap empties the full t... | Mathematics MCQ | Examoogle

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MathematicsTime and Work

A tap fills a tank in 15 hours. Another tap empties the full tank in 30 hours. How long (in hours) will it take to fill the tank completely if both taps are open together?

Correct Answer

📐 MA • Math LCM Efficiency MethodMathematicsTime and Work

Option C

Quick Summary:

Use the product/difference formula: $\frac{x \times y}{y - x}$ . Here, $\frac{15 \times 30}{30 - 15} = \frac{450}{15} = 30$ .

📐MAMath SolutionLCM Efficiency Method

📋 Given

Tap A fills the tank in 15 hours. Tap B empties the full tank in 30 hours.

🔢 Formula Used

Time=Total WorkNet Efficiency=WEA−EB\text{Time} = \frac{\text{Total Work}}{\text{Net Efficiency}} = \frac{W}{E_A - E_B}Time=Net EfficiencyTotal Work​=EA​−EB​W​

⚡ Exam Hall Shortcut / Speed Trick

Use the product/difference formula: $\frac{x \times y}{y - x}$ . Here, $\frac{15 \times 30}{30 - 15} = \frac{450}{15} = 30$ .

⚠️ Common Student Trap / Pitfall

Students often add the rates ( $1/15 + 1/30 = 3/30 = 1/10$ , leading to 10 hours) instead of subtracting the emptying rate from the filling rate.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Calculate Total Capacity

Assume the total capacity of the tank is the LCM of 15 and 30, which is 30 units.

Total Capacity=LCM(15,30)=30 units\text{Total Capacity} = \text{LCM}(15, 30) = 30 \text{ units}Total Capacity=LCM(15,30)=30 units

Determine Individual Rates

The filling rate of Tap A is $30/15 = 2$ units/hr and the emptying rate of Tap B is $30/30 = 1$ unit/hr.

EA=2,EB=1E_A = 2, \quad E_B = 1EA​=2,EB​=1

Calculate Net Efficiency

When both taps are open, the net rate of work is the difference between filling and emptying.

Net Rate=2−1=1 unit/hr\text{Net Rate} = 2 - 1 = 1 \text{ unit/hr}Net Rate=2−1=1 unit/hr

Find Final Time

The total time required to fill the tank is the total capacity divided by the net efficiency.

Time=301=30 hours\text{Time} = \frac{30}{1} = 30 \text{ hours}Time=130​=30 hours

✅

C is correct because the net filling rate of 1 unit per hour means the 30-unit tank will be full in 30 hours.

Core Concepts Used

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Work-Efficiency Principle Pipes and Cisterns Relative Rates

💡 EXAM TIP

This concept of 'Net Efficiency' is identical to relative speed in Time, Speed, and Distance problems where two objects move in opposite directions.

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