You would like to have $$ \$ 5000 $$ in 5 years for a special vacation folowing graduation by making deposits at the end of every six monilhs in an annuty that pays $$ 6 \% $$ compounded semiannually Answer parts (a) and (b) (i) Cick the icon to view some finance formulas a. How much should you deposit at the end of every six montis? In order to have $$ \$ 5000 $$ in $$ \mathbf{5} $$ years, you should deposit $$ \$ 437 $$ at the end of every six months (Do not round untal the final answer. Then round up to the nearest dolat.) D. How much of the $$ \$ 5000 $$ comas from deposits and how riuch comes from earnings? $$ \$ $$ $$ \square $$ of the $$ \$ 5000 $$ comes from your deposits and $$ \$ $$ $$ \square $$ comes from interest (Use the answer from part a to lind this answor. Round to he nearest dollar as needed)

Question

UpStudy AI · Accepted Answer

We are given that you want a future value of  
$$
\$5000
$$  
in $$5$$ years by making deposits at the end of every six months into an annuity with a nominal interest rate of $$6\%$$ compounded semiannually. This means the interest rate per six‐month period is  
$$
i = \frac{6\%}{2} = 3\% = 0.03.
$$
Since deposits are made every six months for $$5$$ years, the total number of payments is  
$$
n = 5 \times 2 = 10.
$$

### Part (a): Find the Deposit Amount

For an ordinary annuity (deposits at the end of each period), the future value is given by the formula  
$$
FV = R \frac{(1+i)^n - 1}{i},
$$
where $$R$$ is the deposit made at the end of each period. We set $$FV = 5000$$ and solve for $$R$$:

$$
5000 = R \frac{(1.03)^{10} - 1}{0.03}.
$$

First, compute $$(1.03)^{10}$$. Using a calculator (or logarithms) we get approximately

$$
(1.03)^{10} \approx 1.343916379.
$$

So,

$$
(1.03)^{10} - 1 \approx 1.343916379 - 1 = 0.343916379.
$$

Now, solve for $$R$$:

$$
R = \frac{5000 \times 0.03}{0.343916379} \approx \frac{150}{0.343916379} \approx 436.91.
$$

The instruction is to round up to the nearest dollar, so

$$
R \approx \$437.
$$

### Part (b): Compute the Contribution from Deposits and Interest

**Total Deposits:**  
Since you make $$10$$ deposits and each deposit (after rounding) is $$\$437$$,

$$
\text{Total Deposits} = 10 \times 437 = \$4370.
$$

**Interest Earned:**  
The interest portion is the difference between the future value and the total deposits:

$$
\text{Interest} = 5000 - 4370 = \$630.
$$

Thus, the breakdown of the $$\$5000$$ is:

- $$\$4370$$ comes from your deposits.
- $$\$630$$ comes from the interest earned.

$$
\boxed{
\begin{array}{l}
\text{(a) Deposit Amount: } \$437 \text{ every six months,} \
\text{(b) Composition: } \$4370 \text{ from deposits and } \$630 \text{ from interest.}
\end{array}
}
$$
To have \$5000 in 5 years with 6% interest compounded semiannually, you should deposit \$437 at the end of every six months. This results in \$4370 from deposits and \$630 from interest.

Upstudy AI Solution

Answer

Solution

Part (a): Find the Deposit Amount

Part (b): Compute the Contribution from Deposits and Interest

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