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Completion requirements
Three friends, Abdul, Sheena, and Jeff, share a paper route. Working
together, it takes \(15\) minutes to deliver all the papers. Abdul can
deliver the papers alone in \(40\) minutes, and Sheena can deliver the
papers alone in \(45\) minutes. How long does it take Jeff to cover the
same route by himself?
Let \(j\) represent the time it takes Jeff to deliver the papers.
LCD: \(360j\)
Verify for \(j \doteq 51 \rm{min}\).
It takes Jeff about \(51\) minutes to deliver the papers by himself.
\[\begin{array}{l}
\frac{1}{{{\rm{time\thinspace taken \thinspace by \thinspace Abdul}}}} + \frac{1}{{{\rm{time \thinspace taken\thinspace by \thinspace Sheena }}}} + \frac{1}{{{\rm{time\thinspace taken \thinspace by\thinspace Jeff }}}} = \frac{1}{{{\rm{time\thinspace taken \thinspace together}}}} \\
\frac{1}{{40}} + \frac{1}{{45}} + \frac{1}{j} = \frac{1}{15}, \thinspace j \ne 0 \\
\end{array}\]
\frac{1}{{{\rm{time\thinspace taken \thinspace by \thinspace Abdul}}}} + \frac{1}{{{\rm{time \thinspace taken\thinspace by \thinspace Sheena }}}} + \frac{1}{{{\rm{time\thinspace taken \thinspace by\thinspace Jeff }}}} = \frac{1}{{{\rm{time\thinspace taken \thinspace together}}}} \\
\frac{1}{{40}} + \frac{1}{{45}} + \frac{1}{j} = \frac{1}{15}, \thinspace j \ne 0 \\
\end{array}\]
LCD: \(360j\)
\[\begin{align}
\left({\frac{1}{{40}}} \right)\left({360j}\right) + \left({\frac{1}{{45}}} \right)\left({360j} \right) + \left({\frac{1}{j}} \right)\left({360j} \right) &= \left( {\frac{1}{{15}}} \right)\left({360j} \right) \\
9j + 8j + 360 &= 24j \\
360 &= 7j \\
\frac{{360}}{7} &= j \\
51.428 &= j \\
51\min &\doteq j \\
\end{align}\]
\left({\frac{1}{{40}}} \right)\left({360j}\right) + \left({\frac{1}{{45}}} \right)\left({360j} \right) + \left({\frac{1}{j}} \right)\left({360j} \right) &= \left( {\frac{1}{{15}}} \right)\left({360j} \right) \\
9j + 8j + 360 &= 24j \\
360 &= 7j \\
\frac{{360}}{7} &= j \\
51.428 &= j \\
51\min &\doteq j \\
\end{align}\]
Verify for \(j \doteq 51 \rm{min}\).
| Left Side | Right Side |
|---|---|
|
\[\begin{array}{r}
\frac{1}{{40}} + \frac{1}{{45}} + \frac{1}{j} \\ \frac{1}{{40}} + \frac{1}{{45}} + \frac{1}{{51}} \\ 0.066... \\ \\ \end{array}\] |
\[\begin{array}{l}
\frac{1}{{15}} \\ 0.066... \\ \end{array}\] |
| LS \(\doteq\) RS\(\hspace{30pt}\) | |
It takes Jeff about \(51\) minutes to deliver the papers by himself.