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Two pipes A and B can fill a tank in 45 ...

Two pipes A and B can fill a tank in 45 hours and 36 hours, respectively. If both the pipes are opened simultaneously, how much time will be taken to fill the tank ?

A

20 hours

B

2 hours

C

10 hours

D

5 hours

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem of how long it will take for two pipes A and B to fill a tank when opened simultaneously, we can follow these steps: ### Step 1: Determine the filling rates of each pipe - **Pipe A** can fill the tank in **45 hours**. Therefore, in one hour, it fills: \[ \text{Rate of Pipe A} = \frac{1 \text{ tank}}{45 \text{ hours}} = \frac{1}{45} \text{ tanks per hour} \] - **Pipe B** can fill the tank in **36 hours**. Therefore, in one hour, it fills: \[ \text{Rate of Pipe B} = \frac{1 \text{ tank}}{36 \text{ hours}} = \frac{1}{36} \text{ tanks per hour} \] ### Step 2: Find a common capacity for the tank To make calculations easier, we can assume the capacity of the tank to be the **Least Common Multiple (LCM)** of the two times (45 hours and 36 hours). - The LCM of 45 and 36 is **180**. Thus, we can assume the tank's capacity is **180 units**. ### Step 3: Calculate the filling rate in units per hour - **Pipe A** fills: \[ \text{Units filled by A in 1 hour} = \frac{180 \text{ units}}{45 \text{ hours}} = 4 \text{ units per hour} \] - **Pipe B** fills: \[ \text{Units filled by B in 1 hour} = \frac{180 \text{ units}}{36 \text{ hours}} = 5 \text{ units per hour} \] ### Step 4: Combine the filling rates When both pipes are opened together, their combined filling rate is: \[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} = 4 \text{ units/hour} + 5 \text{ units/hour} = 9 \text{ units/hour} \] ### Step 5: Calculate the total time to fill the tank Now, to find the total time taken to fill the tank, we use the formula: \[ \text{Time} = \frac{\text{Total Capacity}}{\text{Combined Rate}} = \frac{180 \text{ units}}{9 \text{ units/hour}} = 20 \text{ hours} \] ### Final Answer Thus, the time taken to fill the tank when both pipes A and B are opened simultaneously is **20 hours**. ---
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