Henry Ernest Dudeney/Puzzles and Curious Problems/35 - The Reapers' Puzzle/Solution

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Puzzles and Curious Problems by Henry Ernest Dudeney: $35$

The Reapers' Puzzle
Three men were to receive $90 \shillings$ for harvesting a field, conditionally upon the work being done in $5$ days.
Jake could do it alone in $9$ days, but as Ben was not as good a workman they were compelled to engage Bill for $2$ days,
in consequence of which Ben got $3 \shillings 9 \oldpence$ less than he would otherwise have received.
How long would it have taken Ben and Bill together to complete the work?


Solution

$9 \tfrac {63} {73}$ days.


Proof

Let $a, b, c$ be the rate of working in fields harvested per day of (respectively) Jake, Ben and Bill.

Let $t_a, t_b, t_c$ be the number of days it would take (respectively) Jake, Ben and Bill to harvest the field alone.

Let $s_a, s_b, s_c$ be the number of shillings per day (respectively) Jake, Ben and Bill earn by harvesting fields.


We have that the rate of remuneration is $90 \shillings$ per field in $5$ days.

That is:

$5 \paren {s_a + s_b + s_c} = 90$

So the rate of remuneration is $18 \shillings$ per field per day.


We have that:

$t_a = 9$

That is:

$a = \dfrac 1 9$

Hence after $5$ days, Jake would have got $\dfrac 5 9$ of the field harvested, and would earn $50 \shillings$.

Hence his pay rate is:

$s_a = 10$

Ben would have then earned $40 \shillings$, but $3 \shillings 9 \oldpence$ of that had to be given to Bill.

So Ben is worth $36 \shillings 3 \oldpence$ for $5$ days' work.

Hence all $90 \shillings$ remuneration is accounted for:

Jake earns $50 \shillings$ for $5$ days' work
Ben earns $36 \shillings 3 \oldpence$ for $5$ days' work
Bill earns $3 \shillings 9 \oldpence$ for $2$ days' work.

Hence we have:

$s_a = \dfrac 1 5 \times 50 = 10$
$s_b = \dfrac 1 5 \times 36 \tfrac 1 4 = \dfrac 1 5 \times \dfrac {145} 4 = \dfrac {29} 4 = 7 \tfrac 1 4$
$s_c = \dfrac 1 2 \times 3 \tfrac 3 4 = \dfrac 1 2 \times \dfrac {15} 4 = \dfrac {15} 8 = 1 \tfrac 1 8$

Hence to earn the full $90 \shillings$ Ben needs to work $90 \div \dfrac {29} 4 = \dfrac {360} {29} = 12 \tfrac {12} {29}$ days.

That is:

$t_b = 12 \tfrac {12} {29}$

and so:

$b = \dfrac {29} {360}$

To earn the full $90 \shillings$ Bill needs to work $90 \div \dfrac {15} 8 = 48$ days.

That is:

$t_c = 48$

and so:

$c = \dfrac 1 {48}$


Now let $t$ be the number of days it will take Bill and Ben to harvest the field together.

So:

\(\ds t \paren {b + c}\) \(=\) \(\ds 1\) that is, both together they can harvest $1$ field in $t$ days
\(\ds \leadsto \ \ \) \(\ds t\) \(=\) \(\ds \dfrac 1 {b + c}\)
\(\ds \) \(=\) \(\ds \dfrac 1 {\tfrac {29} {360} + \tfrac 1 {48} }\)
\(\ds \) \(=\) \(\ds \dfrac 1 {\tfrac {58} {720} + \tfrac {15} {720} }\)
\(\ds \) \(=\) \(\ds \dfrac {720} {58 + 15}\)
\(\ds \) \(=\) \(\ds \dfrac {720} {73}\)
\(\ds \) \(=\) \(\ds 9 \tfrac {63} {73}\)

$\blacksquare$


Sources

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