Let $A$ be the balance in a fund on January $1,2023,$ B be the balance in the fund on May $31,2023,$ and $C$ be the balance in the fund on December $31,2023.$

a$)$ If there are no deposits or withdrawals during $2023,$ show that the dollar$-$weighted rate of return and the time$-$weighted rate of return for the fund during $2023$ are both equal to $($C $-$A$)/$A$.$

b$)$ If there was a single deposit of $W$ immediately after the May $31,2023$ balance of $B$ was determined, find expressions for the time$-$weighted rate of return and the dollar$-$weighted rate of return for the portfolio during $2023($assume simple interest for periods less than one year$).$