Henry Ernest Dudeney/Puzzles and Curious Problems/139 - Working Alone/Solution

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Puzzles and Curious Problems by Henry Ernest Dudeney: $139$

Working Alone
Alfred and Bill together can do a job of work in $24$ days.
If Alfred can do two-thirds as much as Bill, how long will it take each of them to do the work alone?


Solution

Alfred would take $60$ days, while Bill would take $40$ days.


Proof

Let $t_a$ and $t_b$ be the time taken for Alfred and Bill to do the job.

Let $a$ and $b$ be the rate in days at which Alfred and Bill can do the job.


In $24$ days, the contributions of Alfred and Bill is $24 a$ and $24 b$ respectively.

So for the total contribution to the job, we have:

\(\ds 24 \paren {a + b}\) \(=\) \(\ds 1\)
\(\ds \leadsto \ \ \) \(\ds 24\) \(=\) \(\ds \dfrac 1 {a + b}\)


We have:

\(\ds b\) \(=\) \(\ds \dfrac 1 {t_b}\)
\(\ds a\) \(=\) \(\ds \dfrac 1 {t_a}\)
\(\ds \) \(=\) \(\ds \dfrac 2 {3 t_b}\)
\(\ds \leadsto \ \ \) \(\ds a + b\) \(=\) \(\ds \dfrac 1 {t_b} + \dfrac 2 {3 t_b}\)
\(\ds \leadsto \ \ \) \(\ds a + b\) \(=\) \(\ds \dfrac 5 {3 t_b}\)


and so:

\(\ds 24\) \(=\) \(\ds \dfrac 1 {a + b}\)
\(\ds \) \(=\) \(\ds \dfrac {3 t_b} 5\)
\(\ds \leadsto \ \ \) \(\ds t_b\) \(=\) \(\ds \dfrac {5 \times 24} 3\)
\(\ds \) \(=\) \(\ds 40\)
\(\ds \leadsto \ \ \) \(\ds t_a\) \(=\) \(\ds \dfrac {3 \times 40} 2\)
\(\ds \) \(=\) \(\ds 60\)

So Alfred would take $60$ days, while Bill would take $40$ days.

$\blacksquare$


Sources

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