Two pipes P and Q would fill an empty cistern in $24$ minutes and $32$ minutes respectively . Both the pipes being opened together , find when the first pipe must be turned off so that the total time taken to fill the empty cistern is $28$ minutes.

Using the LCM method, let the total capacity be 96 units, meaning pipe P fills at 4 units per minute and pipe Q fills at 3 units per minute. Since the total time to fill the cistern is 28 minutes, pipe Q works for the entire duration and fills 84 units. The remaining 12 units must be filled by pipe P, which at 4 units per minute takes exactly 12 minutes before it is turned off.

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