2007 Alabama ARML TST Problems/Problem 3

Problem

Back in 1802, the town idiot of Boratsville, Thungar the Stenchiﬁed, bought some apples from a vendor at the market. He paid 12 cents for the apples, but when he saw how small the apples were, he demanded two extra apples for free, and the vendor agreed. Ultimately, the price Thungar paid was one penny less per dozen apples than he would have paid had he not received the extra apples. How many apples did Thungar purchase (including the two extra)?

Solution

Let's say he took $a$ apples. $a$ apples for 12 cents is $\frac{144}{a}$ cents a dozen. $a+2$ apples for 12 cents is $\frac{144}{a+2}$ cents a dozen. Their difference is 1, so we have

$\frac{144}{a+2}+1=\frac{144}{a}$

$144a+a^2+2a=144a+288$

$a^2+2a-288=0$

$a=\frac{-2\pm\sqrt{4+1152}}{2}=\frac{-2\pm 34}{2}=16$ .

Therefore he bought $\boxed{18}$ apples total.

2007 Alabama ARML TST (Problems)
Preceded by: Problem 2	Followed by: Problem 4
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2007 Alabama ARML TST Problems/Problem 3

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