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

by : $35$

 * 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: