Question

What is the future value of the 10% savings from earnings of $187.45 if it earns 5% annual interest, compounded monthly for 30 years?

Use the compound interest formula to estimate the present value.

A=P(1+rn)nt


(1 point)
Responses

$27.68

$27.68

$37.27

$37.27

$58.75

$58.75

$83.75

Answer 1

To find the future value of the savings from earnings of $187.45 at a 10% savings rate, we first calculate how much money is saved:

Savings = 10% of $187.45
= 0.10 * $187.45
= $18.745

Next, we will use the compound interest formula to calculate the future value (FV) of this savings amount, compounded monthly over 30 years. The compound interest formula is given as:

\[
FV = P \left(1 + \frac{r}{n}\right)^{nt}
\]

Where:
- \( P \) = principal amount (initial investment)
- \( r \) = annual interest rate (decimal)
- \( n \) = number of times that interest is compounded per year
- \( t \) = number of years the money is invested or borrowed for

In this case:

- \( P = 18.745 \)
- \( r = 0.05 \) (5% expressed as a decimal)
- \( n = 12 \) (compounded monthly)
- \( t = 30 \)

Plugging in these values, we get:

\[
FV = 18.745 \left(1 + \frac{0.05}{12}\right)^{12 \times 30}
\]

Calculating the components step by step:

1. Calculate \( \frac{r}{n} \):

\[
\frac{0.05}{12} = 0.0041667
\]

2. Calculate \( nt \):

\[
12 \times 30 = 360
\]

3. Calculate the compound growth factor:

\[
1 + 0.0041667 \approx 1.0041667
\]

4. Raise this to the power of \( nt \):

\[
(1.0041667)^{360} \approx 4.46774
\]

5. Now calculate the future value:

\[
FV = 18.745 \times 4.46774 \approx 83.7472
\]

So, the estimated future value of the savings is approximately **$83.75**.

Thus, the correct response is:

**$83.75**