You would like to have $$ \$ 4,500 $$ in 3 years for a special vacation following graduation by making deposits at the end of every six months in an annuity that pays $$ 3.5 \% $$ compounded semiannually. a. Determine how much you should deposit at the end of every six months. b. How much of the $$ \$ 4,500 $$ comes from deposits and how much comes from interest? a. In order to have $$ \$ 4,500 $$ in 3 years, you should deposit $$ \$ \square $$ at the end of every six months. (Do not round until the final answer. Then round up to the nearest dollar.) b. $$ \$ \square $$ of the $$ \$ 4,500 $$ comes from your deposits and $$ \$ \square $$ comes from interest. (Use the answer from part a to find this answer. Round to the nearest dollar as needed.)

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To determine how much you should deposit every six months to have \$4,500 in 3 years and to understand the composition of that amount, we'll break the problem into two parts:

### **Part a: Calculating the Semiannual Deposit**

**Given:**
- **Future Value (FV):** \$4,500
- **Annual Interest Rate:** 3.5% (compounded semiannually)
- **Time Period:** 3 years
- **Number of Compounding Periods per Year (m):** 2
- **Total Number of Periods (n):** $$ 3 \text{ years} \times 2 = 6 $$ periods
- **Periodic Interest Rate (r):** $$ \frac{3.5\%}{2} = 1.75\% $$ or 0.0175 in decimal

**Formula for the Future Value of an Ordinary Annuity:**
$$
FV = PMT \times \left( \frac{(1 + r)^n - 1}{r} \right)
$$
Where:
- $$ PMT $$ = payment made each period

**Rearranging the Formula to Solve for $$ PMT $$:**
$$
PMT = \frac{FV}{\left( \frac{(1 + r)^n - 1}{r} \right)}
$$

**Plugging in the Values:**
$$
PMT = \frac{4500}{\left( \frac{(1 + 0.0175)^6 - 1}{0.0175} \right)}
$$

**Calculating $$ (1 + 0.0175)^6 $$:**
$$
(1.0175)^6 \approx 1.1097
$$

**Substituting Back:**
$$
PMT = \frac{4500}{\left( \frac{1.1097 - 1}{0.0175} \right)} = \frac{4500}{6.2744} \approx 717.2
$$

**Rounding Up:**
$$
PMT \approx \$718
$$

**Answer to Part a:**
$$
\boxed{\$718}
$$

---

### **Part b: Determining the Composition of the \$4,500**

**Total Deposits Made:**
$$
\text{Total Deposits} = PMT \times n = 718 \times 6 = 4,308
$$

**Interest Earned:**
$$
\text{Interest} = FV - \text{Total Deposits} = 4,500 - 4,308 = 192
$$

**Answer to Part b:**
$$
\boxed{\$4,308 \text{ from deposits and } \$192 \text{ from interest}}
$$
You should deposit \$718 at the end of every six months. Of the \$4,500, \$4,308 comes from your deposits and \$192 comes from interest.

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