Henry Ernest Dudeney/Puzzles and Curious Problems/15 - Buying Turkeys/Solution

From ProofWiki
Jump to navigation Jump to search

Puzzles and Curious Problems by Henry Ernest Dudeney: $15$

Buying Turkeys
A man bought a number of turkeys at a cost of $\pounds 60$,
and after reserving fifteen of the birds he sold the remainder for $\pounds 54$,
thus gaining $2 \shillings$ a head by these.
How may turkeys did he buy?


Solution

$75$ turkeys for $16 \shillings$ each.

He sold $60$ turkeys for $18 \shillings$ each.


Proof

Let $n$ be the number of turkeys bought.

Let $t_1 \shillings$ be the price he paid per turkey for the original $n$ turkeys.

Let $t_2 \shillings$ be the price he sold the remaining $n - 15$ turkeys for.

All monetary values will be expressed in shillings.

We have:

\(\ds n t_1\) \(=\) \(\ds 60 \times 20\) A man bought a number of turkeys at a cost of $\pounds 60$,
\(\ds \paren {n - 15} t_2\) \(=\) \(\ds 54 \times 20\) and after reserving fifteen of the birds he sold the remainder for $\pounds 54$,
\(\ds t_2\) \(=\) \(\ds t_1 + 2\) thus gaining $2 \shillings$ a head by these.
\(\ds \leadsto \ \ \) \(\ds \paren {n - 15} \paren {t_1 + 2}\) \(=\) \(\ds 54 \times 20\) eliminating $t_2$
\(\ds \leadsto \ \ \) \(\ds n t_1 + 2 n - 15 t_1 - 30\) \(=\) \(\ds 54 \times 20\) simplifying
\(\ds \leadsto \ \ \) \(\ds 60 \times 20 + 2 n - 15 \times \dfrac {60 \times 20} n - 30\) \(=\) \(\ds 54 \times 20\) as $n t_1 = 60 \times 20$
\(\ds \leadsto \ \ \) \(\ds n^2 + 45 n - 9000\) \(=\) \(\ds 0\) simplifying
\(\ds \leadsto \ \ \) \(\ds n\) \(=\) \(\ds \dfrac {-45 \pm \sqrt {45^2 + 4 \times 9000} } 2\) Quadratic Formula
\(\ds \) \(=\) \(\ds \dfrac {-45 \pm 195} 2\) calculating
\(\ds \) \(=\) \(\ds 75 \text { or } -120\)

Clearly only the positive solution makes sense.

$\blacksquare$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Henry_Ernest_Dudeney/Puzzles_and_Curious_Problems/15_-_Buying_Turkeys/Solution&oldid=561450"