Let $f(a) = \frac{13+a}{3a+7}$ where $a$ is restricted to positive integers. What is the maximum value of $f(a)$? Let $f(a) = \frac{13+a}{3a+7}$ where $a$ is restricted to positive integers. What is the maximum value of $f(a)$?
I tried graphing but it didn't help.  Could anyone answer?  Thanks!

\begin{align}
   f(a)
   &= \dfrac{a+13}{3a+7} \\
   &= \dfrac 13 \dfrac{3a+39}{3a+7} \\
   &= \dfrac 13 \left(1 + \dfrac{32}{3a+7} \right) \\
\end{align}
This implies that $f(a)$ is strictly decreasing for positive integers.
Hence the maximum value must be $f(1) = \dfrac 75$
