Walt and Cady Longmire were investing in shares of Wyoming Mining Company. The returns for the stocks during $2017,2018,$ and $2019$ were $10\%,-5\%,$ and $23\mathrm{.}5\%,$ respectively. At the beginning of $2017,$ Walt invested $$5,000$ in the stocks and he did not make any additional investment in $2018.$ At the end of $2018,$ Walt cashed out his entire investment in Wyoming Mining Company.

Meanwhile, his daughter Cady Longmire invested $$3,000$ in the stocks at the beginning of $2017.$ She later made no deposit in $2018$ but invested an additional $$5,000$ at the start of $2019.$ At the end of $2019,$ Cady cashed out her entire investment in Wyoming Mining Company.

Answer the following:

$($i$)$ Calculate and interpret Walt$$s dollar$-$weighted average return for his $2-$year investment in Wyoming Mining Company.

$($ii$)$ Compare Walt$$s dollar$-$weighted average return to the geometric average return $($for the $2-$year investment period$),$ do your answers make sense? Why or Why not? Did you have any expectation prior to your calculation?

$($iii$)$ Calculate and interpret Cady$$s dollar$-$weighted average return for her $3-$year investment in Wyoming Mining Company.

$($iv$)$ Compare Cady$$s dollar$-$weighted average return to the geometric average return $($for the $3-$year investment period$),$ do your answers make sense? Why or Why not? Did you have any expectation prior to your calculation?