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Lisa​, an experienced shipping​ clerk, can fill a certain order in 12 hours. Bill​, a new​ clerk, needs 13 hours to do the same job. Working​ together, how long will it take them to fill the​ order?

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To determine how long it will take Lisa and Bill to fill an order together, we can find their individual work rates and then combine them.

  1. Calculate their individual work rates:

    • Lisa can complete the order in 12 hours, so her work rate is \( \frac{1}{12} \) of the order per hour.
    • Bill can complete the order in 13 hours, so his work rate is \( \frac{1}{13} \) of the order per hour.

  2. Add their work rates together: \[ \text{Combined work rate} = \frac{1}{12} + \frac{1}{13} \]

    To add these fractions, we need a common denominator. The least common multiple (LCM) of 12 and 13 is 156.

    • Convert \( \frac{1}{12} \) to have a denominator of 156: \[ \frac{1}{12} = \frac{13}{156} \]
    • Convert \( \frac{1}{13} \) to have a denominator of 156: \[ \frac{1}{13} = \frac{12}{156} \]

    Now, add the two fractions: \[ \text{Combined work rate} = \frac{13}{156} + \frac{12}{156} = \frac{25}{156} \]

  3. Find the time taken to complete one order working together: If their combined work rate is \( \frac{25}{156} \) of the order per hour, then the time \( t \) in hours that they will take to complete one whole order is the reciprocal of their combined work rate: \[ t = \frac{1}{\text{Combined work rate}} = \frac{1}{\frac{25}{156}} = \frac{156}{25} \]

  4. Calculate \( \frac{156}{25} \): \[ t = 6.24 \text{ hours} \]

Thus, working together, Lisa and Bill will take approximately 6.24 hours to fill the order. In hours and minutes, this is about 6 hours and 15 minutes.