Suppose a two digit whole number is divided by the sum of its digits, what are the largest and smallest possible values? So we can write a two digit whole number as $n = 10a+b$ where $1 \leq a,b \leq 10$ and we would have that we want to minimize/maximize the following functions:
$f(a,b) = 10a+b$
$g(a,b) = a+b$
I don't remember how one does this and I don't know if there is another approach that could work.
Write the ratio you are interested in $$\frac {10a+b}{a+b}=1+\frac {9a}{a+b}=10-\frac {9b}{a+b}$$ To make this large you want $a$ large and $b$ small. If you allow $b=0$ this becomes $10$ regardless of $a$. If you do not allow $b=0$ the maximum comes at $91$ with $\frac {91}{10}=9.1$. To make it small you want $b$ large and $a$ small, but we cannot have $a=0$, so the minimum is $19$ with $\frac {19}{10}=1.9$