Sure! Let’s break down the given question into parts (a), (b), and (c) and find the relevant solutions for each part.
### Part (a):
We need to find the future value of an annuity where the woman is depositing [tex]$\$[/tex]14,000[tex]$ at the end of each year for 13 years in an account that has an interest rate of \(5\%\) compounded annually.
The formula for the future value of an ordinary annuity is:
\[
FV = PMT \left[\frac{(1 + i)^n - 1}{i}\right]
\]
where:
- \(PMT\) is the annual payment (deposit),
- \(i\) is the annual interest rate, and
- \(n\) is the number of periods (years).
Substitute the given values into the formula:
- \(PMT = 14,000\),
- \(i = 0.05\),
- \(n = 13\).
\[
FV = 14000 \left[\frac{(1 + 0.05)^{13} - 1}{0.05}\right]
\]
After evaluating this expression, we find:
\[
FV = \$[/tex]247,981.76
\]
Thus, the value in the account at the end of 13 years will be [tex]$\$[/tex]247,981.76[tex]$.
### Part (b):
Now, we need to find the future value if the woman deposits the same amount (\$[/tex]14,000) at the end of each year for 13 years, but this time the investment firm pays [tex]\(4\%\)[/tex] compounded annually.
Using the same future value formula for an ordinary annuity:
[tex]\[
FV = PMT \left[\frac{(1 + i)^n - 1}{i}\right]
\][/tex]
Substitute the new interest rate:
- [tex]\(PMT = 14,000\)[/tex],
- [tex]\(i = 0.04\)[/tex],
- [tex]\(n = 13\)[/tex].
[tex]\[
FV = 14000 \left[\frac{(1 + 0.04)^{13} - 1}{0.04}\right]
\][/tex]
After evaluating this expression, we find:
[tex]\[
FV = \$232,775.73
\][/tex]
Therefore, at her sister’s firm, she will have [tex]$\$[/tex]232,775.73[tex]$ in her account at the end of 13 years.
### Part (c):
Finally, we need to calculate the gain or loss if the woman invests in her sister’s firm compared to the first investment firm.
The gain or loss is determined by:
\[
\text{Gain/Loss} = FV_{\text{sister's firm}} - FV_{\text{first firm}}
\]
From parts (a) and (b):
- \(FV_{\text{first firm}} = \$[/tex]247,981.76\),
- [tex]\(FV_{\text{sister's firm}} = \$232,775.73\)[/tex].
Therefore:
[tex]\[
\text{Gain/Loss} = 232,775.73 - 247,981.76 = -\$15,206.03
\][/tex]
This means she would lose [tex]$\$[/tex]15,206.03[tex]$ over the 13 years by investing in her sister’s firm compared to the first investment firm.
Conclusively, her outcomes are:
(a) Value in the first firm after 13 years: $[/tex]247,981.76[tex]$.
(b) Value in her sister's firm after 13 years: $[/tex]232,775.73[tex]$.
(c) Loss over 13 years by investing in her sister's firm: $[/tex]15,206.03$.
