Henry Ernest Dudeney/Modern Puzzles/19 - Market Transactions

by : $19$

 * Market Transactions
 * A farmer goes to market and buys $100$ animals at a total cost of $\pounds 100$.
 * The price of cows being $\pounds 5$ each,
 * sheep $\pounds 1$ each,
 * and rabbits $1 \shillings$ each,
 * how many of each kind does he buy?


 * Most people will solve this, if they succeed at all, by more or less laborious trial,
 * but there are several direct ways of getting the solution.

Solution

 * $19$ cows, $1$ sheep and $80$ rabbits.

Proof
Recall:
 * $1$ pound sterling ($\pounds 1$) is $20$ shilling ($20 \shillings$)

Let all prices be expressed, therefore, in shillings.

Thus:


 * the price of cows is $5 \times 20 = 100 \shillings$ each
 * the price of sheep is $20 \shillings$ each
 * the price of rabbits is $1 \shillings$ each

and:
 * the total amount of money to spend is $100 \times 20 = 2000 \shillings$.

Let $c$, $s$ and $r$ denote the number of cows, sheep and rabbits respectively.

We have:

Note that both $c$ and $s$ need to be positive.

We need to find possible values of $c$ such that $1900 - 99 c$ is divisible by $19$.

This can happen only when $c$ itself is divisible by $19$.

It is implicit that there are at least some cows are bought, so the solution:
 * $c = 0, s = 100, r = 0$

is usually ruled out.

Hence we have:
 * $c = 19, s = 1, r = 80$