Working alone, Mary can pave a driveway in \(8\) hours and Hillary can pave the same driveway in \(6\) hours. When they

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Sat Aug 05, 2023 7:49 am

Working alone, Mary can pave a driveway in \(8\) hours and Hillary can pave the same driveway in \(6\) hours. When they work together, Mary thrives on teamwork so her rate increases by \(33.33\%\), but Hillary becomes distracted and her rate decreases by \(50\%\). If they both work together, how many hours will it take to pave the driveway?

A. \(3\) hours
B. \(4\) hours
C. \(5\) hours
D. \(6\) hours
E. \(7\) hours

OA B

We have:

Individual rates:

\(\bullet\) Mary \(= \dfrac{1}{8}\) per hour

\(\bullet\) Hillary \(= \dfrac{1}{6}\) per hour

Rate when working together:

\(\bullet\) Mary \(= \dfrac{1}{8} + \left(\dfrac{1}{3} \cdot \dfrac{1}{8}\right) = \dfrac{3}{24} + \dfrac{1}{24} = \dfrac{4}{24} = \dfrac{1}{6}\) per hour

\(\bullet\) Hillary \(= \dfrac{1}{6} - \left(\dfrac{1}{2} \cdot \dfrac{1}{6}\right) = \dfrac{2}{12} - \dfrac{1}{12} = \dfrac{1}{12}\) per hour

Together they work:

\(\bullet\) \(\dfrac{1}{6} + \dfrac{1}{12} = \dfrac{2}{12} + \dfrac{1}{12} = \dfrac{3}{12} = \dfrac{1}{4}\) per hour

So they will need \(4\) hours to complete the driveway.

Therefore, B