One pipe can fill a cistern in $3\ hours$ less than the other. The two pipes together can fill the cistern in $6\ hours$ $40\ minutes$. Find the time that each pipe will take to fill the cistern.

Using the combined work rate formula, their combined rate is one divided by six and two-thirds hours, which is three twentieths per hour. Assuming the faster pipe takes fifteen hours and the slower pipe takes twelve hours, their individual rates are one fifteenth and one twelfth per hour. Adding one fifteenth and one twelfth gives nine sixtieths, which simplifies to three twentieths per hour, perfectly matching the required combined rate. Therefore, the individual times are twelve hours and fifteen hours.

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