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Three taps A,B and C together can fill a...

Three taps A,B and C together can fill an empty tank in 10 minutes. The tap A alone can fill it in 30 minutes and the tap B alone in 40 minutes. How long will the tap C alone take to fill it ?

A

16 minutes

B

24 minutes

C

32 minutes

D

40 minutes

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem step by step, we can follow these calculations: ### Step 1: Determine the rates of filling for each tap - Tap A can fill the tank in 30 minutes. Therefore, the rate of tap A is: \[ \text{Rate of A} = \frac{1 \text{ tank}}{30 \text{ minutes}} = \frac{1}{30} \text{ tanks per minute} \] - Tap B can fill the tank in 40 minutes. Therefore, the rate of tap B is: \[ \text{Rate of B} = \frac{1 \text{ tank}}{40 \text{ minutes}} = \frac{1}{40} \text{ tanks per minute} \] - Together, taps A, B, and C can fill the tank in 10 minutes. Therefore, their combined rate is: \[ \text{Rate of A + B + C} = \frac{1 \text{ tank}}{10 \text{ minutes}} = \frac{1}{10} \text{ tanks per minute} \] ### Step 2: Set up the equation for tap C Let the rate of tap C be \( \text{Rate of C} = \frac{1}{x} \) tanks per minute, where \( x \) is the time taken by tap C alone to fill the tank. From the rates, we have: \[ \text{Rate of A} + \text{Rate of B} + \text{Rate of C} = \text{Rate of A + B + C} \] Substituting the known rates: \[ \frac{1}{30} + \frac{1}{40} + \frac{1}{x} = \frac{1}{10} \] ### Step 3: Find a common denominator and solve for \( x \) The least common multiple (LCM) of 30, 40, and 10 is 120. We will multiply through by 120 to eliminate the fractions: \[ 120 \left( \frac{1}{30} \right) + 120 \left( \frac{1}{40} \right) + 120 \left( \frac{1}{x} \right) = 120 \left( \frac{1}{10} \right) \] This simplifies to: \[ 4 + 3 + \frac{120}{x} = 12 \] Combining the constants: \[ 7 + \frac{120}{x} = 12 \] Now, isolate \( \frac{120}{x} \): \[ \frac{120}{x} = 12 - 7 \] \[ \frac{120}{x} = 5 \] ### Step 4: Solve for \( x \) Cross-multiplying gives: \[ 120 = 5x \] Dividing both sides by 5: \[ x = \frac{120}{5} = 24 \] ### Conclusion Thus, tap C alone will take **24 minutes** to fill the tank. ---
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