2310.11 – Fleet Street Market

At the Fleet Street Market, Kayla sells a certain fixed number of free-range geese for $90. If you want free-range chickens, $140 buys you four more chickens than geese. Oh, and six chickens and two geese will cost you $150. How much does one chicken cost? How much for one goose?

Solution

Let $n$ = “that certain number of geese that Kayla sells.” Let $g$ = the price of one goose and let $c$ = the price of one chicken. Then we know that,

(1) $ng = 90,$

(2) $(n + 4)c = 140,$

(3) $2g + 6c = 150.$

From (1) we get, $g=90/n$ ; from (2) we get $c=140/(n+4)$ . We now put everything in terms of $n$ by substituting into (3).

$2 \frac{90}{n} + 6 \frac{140}{n+4}=150.$

Continuing,

\begin{aligned} \frac{180}{n} + \frac{840}{n+4} &=& 150, \\ \frac{6}{n} + \frac{28}{n+4} &=& 5, \\ 6 (n+4) + 28 n &=& 5 n (n+4), \\ 5 n^2 - 14 n - 24 &=& 0, \\ (5n + 6) (n - 4) &=& 0. \end{aligned}

Rejecting the negative root ( $-6/5$ ), we may conclude that $n = 4$ , $g = 90/4 = 22.50$ and $c = 17.5$ . So, a chicken costs $17.50 and a goose costs $22.50.