CEMC Fermat

Let $n$ be the number of cookies in the cookie jar. Let $r$ be the number of raisins in each of the $n-1$ smaller, identical cookies. This means that there are $r+1$ raisins in the larger cookie. If we removed one raisin from the larger cookie, it too would have $r$ raisins and so each of the $n$ cookies would have the same number of raisins $(r)$, and the total number of raisins in the cookies would be $100-1=99$. From this, we obtain $nr=99$. Since $n$ is between 5 and 10, then we must have $99=9 \times 11$; that is, $n=9$ and $r=11$. Since there are 11 raisins in each of the smaller cookies, then there are $11+1=12$ raisins in the larger cookie.