Two pipes A & B can fill a cistern in 8 and 12 minutes respectively. If both the pipes are opened simultaneously, then after how much time should B be closed so that the tank is filled in 6 minutes?
Question
Two pipes A & B can fill a cistern in 8 and 12 minutes respectively. If both the pipes are opened simultaneously, then after how much time should B be closed so that the tank is filled in 6 minutes?
Solution
Sure, let's solve this step by step:
Step 1: First, we need to find out how much of the tank each pipe can fill in one minute.
Pipe A can fill the tank in 8 minutes, so in one minute it fills 1/8 of the tank. Pipe B can fill the tank in 12 minutes, so in one minute it fills 1/12 of the tank.
Step 2: Now, let's find out how much of the tank both pipes can fill together in one minute.
If we add the fractions we got in step 1, we get (1/8 + 1/12) = 5/24. So, together, pipes A and B can fill 5/24 of the tank in one minute.
Step 3: We know that the tank needs to be filled in 6 minutes. If both pipes are open for the entire 6 minutes, they would fill (5/24 * 6) = 5/4 = 1.25 tanks, which is more than we need.
Step 4: Therefore, we need to close pipe B at some point. Let's say we need to close pipe B after x minutes.
In these x minutes, pipe A would fill (1/8 * x) of the tank and pipe B would fill (1/12 * x) of the tank.
Step 5: After x minutes, we close pipe B and let pipe A continue to fill the tank for the remaining (6 - x) minutes. In this time, pipe A would fill an additional (1/8 * (6 - x)) of the tank.
Step 6: The total amount of the tank filled must be 1 (the whole tank). So, we can set up the equation:
(1/8 * x) + (1/12 * x) + (1/8 * (6 - x)) = 1
Step 7: Solving this equation for x gives us x = 4.8 minutes.
So, pipe B should be closed after 4.8 minutes.
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