Answer
$\dfrac{r-s}{r}$
Work Step by Step
Expressing the numerators and the denominators as similar fractions, the given expression, $
\dfrac{\dfrac{r}{s}-\dfrac{s}{r}}{\dfrac{r}{s}+1}
,$ is equivalent to
\begin{align*}
&
\dfrac{\dfrac{r}{s}\cdot\dfrac{r}{r}-\dfrac{s}{r}\cdot\dfrac{s}{s}}{\dfrac{r}{s}+\dfrac{s}{s}}
\\\\&=
\dfrac{\dfrac{r^2}{rs}-\dfrac{s^2}{rs}}{\dfrac{r+s}{s}}
\\\\&=
\dfrac{\dfrac{r^2-s^2}{rs}}{\dfrac{r+s}{s}}
\\\\&=
\dfrac{r^2-s^2}{rs}\div\dfrac{r+s}{s}
.\end{align*}
Using $a^2-b^2=(a+b)(a-b)$, the factored form of the expression above is
\begin{align*}
&
\dfrac{(r+s)(r-s)}{rs}\div\dfrac{r+s}{s}
&
.\end{align*}
Multiplying by the reciprocal of the divisor and cancelling the common factor/s, the expression above is equivalent to
\begin{align*}\require{cancel}
&
\dfrac{(r+s)(r-s)}{rs}\cdot\dfrac{s}{r+s}
\\\\&=
\dfrac{\cancel{(r+s)}(r-s)}{r\cancel s}\cdot\dfrac{\cancel s}{\cancel{r+s}}
\\\\&=
\dfrac{r-s}{r}
.\end{align*}
Hence, the expression $
\dfrac{\dfrac{r}{s}-\dfrac{s}{r}}{\dfrac{r}{s}+1}
$ simplifies to $
\dfrac{r-s}{r}
$.
