2017 AMC 8
Complete problem set with solutions and individual problem pages
Starting with some gold coins and some empty treasure chests, I tried to put gold coins in each treasure chest, but that left treasure chests empty. So instead I put gold coins in each treasure chest, but then I had gold coins left over. How many gold coins did I have?
- A.
- B.
- C.
- D.
- E.
Solution 1
We can represent the amount of gold with and the amount of chests with . We can use the problem to make the following equations:
We do this because for chests there are empty and if were in each multiplied by is left.
Therefore, This implies that We therefore have So, our answer is .
Solution 2
With coins, there are chests, by the first condition. These don't fit in with the second condition, so we move onto coins. By the same first condition, there are chests(). This also doesn't fit with the second condition. So, onto coins. The first condition implies that there are chests, which DOES fit with the second condition, since . Thus, the desired value is .
