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,

$\begin{equation}ng = 90\end{equation}$

$\begin{equation}(n + 4)c = 140\end{equation}$

$\begin{equation}2g + 6c = 150\end{equation}$

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

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

Continuing,

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

Rejecting the negative root ( $\dfrac{-6}{5}$ ), 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.