Stefani German, a $40-$year$-$old woman, plans to retire at age $65,$ and she wants to accumulate $$400,000$ over the next $25$ years to supplement the retirement programs provided by the federal government and her employer. She expects to earn an average annual return of about $5\%$ by investing in a low$-$risk portfolio containing about $20\%$ short$-$term securities$,30\%$ common stock, and $50\%$ bonds.

Stefani currently has $$41,342$ that at an annual rate of return of $5\%$ will grow to about $$140,000$ by her $65$ th birthday $($the $$140,000$ figure is found using time value of money techniques, Chapter $4$ Appendix.$)$ Stefani consults a financial advisor to determine how much money she should save each year to meet her retirement savings objective. The advisor tells Stefani that if she saves about $$20\mathrm{.}95$ each year, she will accumulate $$1,000$ by age $65.$ Saving $5$ times that amount each year, $$104\mathrm{.}75,$ allows Stefani to accumulate roughly $$5,000$ by age $65.$

a$.$ How much additional money does Stefani need to accumulate over time to reach her goal of $$400,000?$

b$.$ How much must Stefani save to accumulate the sum calculated in part a over the next $25$ years?