Question

Veronica plans to make a $95 a month annuity payment to an account that earns 3% annual interest to build up her savings. How much can she save in 10 years with this plan if the interest is compounded monthly?

Use S=P((1+rn)nt−1rn)
.

(1 point)
Responses

$7,122.49

$7,122.49

$13,275.43

$13,275.43

$21,846.27

$21,846.27

$38,960.76

Answer 1

To calculate the future value of an annuity with monthly payments, we can use the formula:

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

Where:
- \( S \) is the future value of the annuity.
- \( P \) is the monthly payment amount.
- \( r \) is the annual interest rate (decimal).
- \( n \) is the number of times that interest is compounded per year.
- \( t \) is the number of years.

Given:
- \( P = 95 \) (the monthly payment)
- \( r = 0.03 \) (3% annual interest)
- \( n = 12 \) (compounded monthly)
- \( t = 10 \) (for 10 years)

First, let's calculate \( r/n \):

\[
\frac{r}{n} = \frac{0.03}{12} = 0.0025
\]

Next, we calculate the total number of payments, \( nt \):

\[
nt = 12 \times 10 = 120
\]

Now we can substitute these values into the formula:

\[
S = 95 \left( \frac{(1 + 0.0025)^{120} - 1}{0.0025} \right)
\]

Calculating \( (1 + 0.0025)^{120} \):

\[
(1 + 0.0025)^{120} \approx 1.349353
\]

Now substituting this into the future value formula:

\[
S = 95 \left( \frac{1.349353 - 1}{0.0025} \right)
\]

Calculating \( 1.349353 - 1 \):

\[
1.349353 - 1 = 0.349353
\]

Now, we size that up:

\[
\frac{0.349353}{0.0025} \approx 139.7412
\]

So now we find \( S \):

\[
S = 95 \times 139.7412 \approx 13,275.43
\]

Therefore, the amount Veronica can save in 10 years with her plan is **$13,275.43**.