1987 AIME Problems/Problem 8

Revision as of 15:26, 15 March 2011 by Pinkmuskrat (talk | contribs) (Solution)

Problem

What is the largest positive integer $n$ for which there is a unique integer $k$ such that $\frac{8}{15} < \frac{n}{n + k} < \frac{7}{13}$?

Solution

Solution 1 Multiplying out all of the denominators, we get:

\begin{align*}104(n+k) &< 195n< 105(n+k)\\ 0 &< 91n - 104k < n + k\end{align*}

Since $91n - 104k < n + k$, $k > \frac{6}{7}n$. Also, $0 < 91n - 104k$, so $k < \frac{7n}{8}$. Thus, $48n < 56k < 49n$. $k$ is unique if it is within a maximum range of $112$, so $n = 112$.

Solution 2 Flip all of the fractions for

$$ (Error compiling LaTeX. Unknown error_msg)\begin{align*}\frac{15}{8} > \frac{k + n}{n} > \frac{13}{7}\

$105n > 56 (k + n) > 104n$ (Error compiling LaTeX. Unknown error_msg)49n > 56k > 48n\end{align*}$$ (Error compiling LaTeX. Unknown error_msg)

Continue as in Solution 1.

See also

1987 AIME (ProblemsAnswer KeyResources)
Preceded by
Problem 7
Followed by
Problem 9
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions