Competition problem — Number theory

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Number theory Difficulty 3.1 AMC 10/12 Find the answer

Starting with some gold coins and some empty treasure chests, I tried to put $9$ gold coins in each treasure chest, but that left $2$ treasure chests empty. So instead I put $6$ gold coins in each treasure chest, but then I had $3$ gold coins left over. How many gold coins did I have?

Pick one

Solution

We can represent the amount of gold with $g$ and the amount of chests with $c$ . We can use the problem to make the following equations:
$9c-18 = g$
$6c+3 = g$
We do this because for 9 chests there are 2 empty and if 9 were in each 9 multiplied by 2 is 18 left.
Therefore, $6c+3 = 9c-18.$ This implies that $c = 7.$ We therefore have $g = 45.$ So, our answer is $\boxed{\textbf{(C)}\ 45}$ .
~CHECKMATE2021

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.