Question: You would like to have $5000 in 5 years for a special vacation following graduation by making deposits at the end of every six months in an annuity that pays 9% compounded semiannually. Answer parts (a) and (b).
Part 1 of 2
a. How much should you deposit at the end of every six months?
In order to have $5000 in 5 years, you should deposit $ enter your response here at the end of every six months.
(Do not round until the final answer. Then round up to the nearest dollar.)
1. **State the problem:**
You want to accumulate $5000 in 5 years by making equal deposits at the end of every six months into an annuity that pays 9% interest compounded semiannually.
2. **Identify the formula:**
The future value of an ordinary annuity is given by:
$$ FV = P \times \frac{(1 + r)^n - 1}{r} $$
where:
- $FV$ is the future value ($5000$),
- $P$ is the payment per period (what we want to find),
- $r$ is the interest rate per period,
- $n$ is the total number of payments.
3. **Calculate parameters:**
- Annual interest rate = $9\% = 0.09$
- Compounded semiannually means 2 periods per year, so
$$ r = \frac{0.09}{2} = 0.045 $$
- Number of years = $5$
- Number of periods = $5 \times 2 = 10$
4. **Plug values into the formula and solve for $P$:**
$$ 5000 = P \times \frac{(1 + 0.045)^{10} - 1}{0.045} $$
Calculate the numerator:
$$ (1 + 0.045)^{10} = 1.045^{10} \approx 1.551328 $$
So:
$$ \frac{1.551328 - 1}{0.045} = \frac{0.551328}{0.045} \approx 12.2517 $$
Therefore:
$$ 5000 = P \times 12.2517 $$
Divide both sides by 12.2517:
$$ P = \frac{5000}{12.2517} $$
Show cancellation:
$$ P = \frac{5000}{\cancel{12.2517}} \approx 408.07 $$
5. **Final answer:**
You should deposit approximately $\boxed{409}$ at the end of every six months (rounded up to the nearest dollar).