Present Value Compound 7Aa5D8

1. **State the problem:** Jane wants to know the present value (initial investment) needed to have $148,700 after 16 years with an interest rate of 3.58% compounded daily. 2. **Formula used:** The compound interest formula for future value is: $$ A = P \left(1 + \frac{r}{n}\right)^{nt} $$ where: - $A$ is the amount of money accumulated after $t$ years, including interest. - $P$ is the principal (the initial amount to find). - $r$ is the annual interest rate (decimal). - $n$ is the number of times interest is compounded per year. - $t$ is the time the money is invested for in years. 3. **Given values:** - $A = 148700$ - $r = 3.58\% = 0.0358$ - $n = 365$ (compounded daily) - $t = 16$ 4. **Rearrange formula to solve for $P$:** $$ P = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}} $$ 5. **Calculate the denominator:** $$ 1 + \frac{0.0358}{365} = 1 + 0.00009808219 = 1.00009808219 $$ 6. **Calculate the exponent:** $$ nt = 365 \times 16 = 5840 $$ 7. **Calculate the compound factor:** $$ \left(1.00009808219\right)^{5840} $$ Using a calculator: $$ \approx 1.747422 $$ 8. **Calculate present value $P$:** $$ P = \frac{148700}{1.747422} \approx 85088.68 $$ 9. **Round to nearest dollar:** $$ P \approx 85089 $$ **Answer:** Jane needs to invest approximately $85089 now to have $148700 after 16 years at 3.58% interest compounded daily.