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 + r/n)^nt -1 / r/n)
Answers below
$7,122.49
$13275.43
$21,846.27
$38,960.76
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 + r/n)^nt -1 / r/n)
Answers below
$7,122.49
$13275.43
$21,846.27
$38,960.76
Answers
GPT-4o mini
To calculate how much Veronica can save with her annuity payments, we can use the formula for the future value of an ordinary annuity:
\[
S = P \left( \frac{(1 + r/n)^{nt} - 1}{r/n} \right)
\]
Where:
- \( S \) is the future value of the annuity,
- \( P \) is the payment amount per period,
- \( r \) is the annual interest rate (as a decimal),
- \( n \) is the number of compounding periods per year,
- \( t \) is the number of years.
For this problem:
- \( P = 95 \) (monthly payment),
- \( r = 0.03 \) (3% annual interest),
- \( n = 12 \) (compounded monthly),
- \( t = 10 \) (10 years).
First, we will calculate \( r/n \):
\[
r/n = \frac{0.03}{12} = 0.0025
\]
Next, we calculate \( nt \):
\[
nt = 12 \times 10 = 120
\]
Now we will substitute these values into the future value formula:
\[
S = 95 \left( \frac{(1 + 0.0025)^{120} - 1}{0.0025} \right)
\]
Calculating \( (1 + 0.0025)^{120} \):
\[
(1 + 0.0025)^{120} = (1.0025)^{120} \approx 1.348850
\]
Now we can substitute this into the formula:
\[
S = 95 \left( \frac{1.348850 - 1}{0.0025} \right)
\]
Calculating the numerator:
\[
1.348850 - 1 = 0.348850
\]
Now we will substitute this back into the equation:
\[
S = 95 \left( \frac{0.348850}{0.0025} \right) = 95 \times 139.54 \approx 13205.36
\]
Finally, let's get the total amount:
\[
S \approx 13205.36
\]
When we round to two decimal places, we find:
\[
S \approx 13205.36
\]
So, looking at the closest answer choices provided, we have \( S \approx 13275.43 \).
Thus, the final answer is:
**$13,275.43**
\[
S = P \left( \frac{(1 + r/n)^{nt} - 1}{r/n} \right)
\]
Where:
- \( S \) is the future value of the annuity,
- \( P \) is the payment amount per period,
- \( r \) is the annual interest rate (as a decimal),
- \( n \) is the number of compounding periods per year,
- \( t \) is the number of years.
For this problem:
- \( P = 95 \) (monthly payment),
- \( r = 0.03 \) (3% annual interest),
- \( n = 12 \) (compounded monthly),
- \( t = 10 \) (10 years).
First, we will calculate \( r/n \):
\[
r/n = \frac{0.03}{12} = 0.0025
\]
Next, we calculate \( nt \):
\[
nt = 12 \times 10 = 120
\]
Now we will substitute these values into the future value formula:
\[
S = 95 \left( \frac{(1 + 0.0025)^{120} - 1}{0.0025} \right)
\]
Calculating \( (1 + 0.0025)^{120} \):
\[
(1 + 0.0025)^{120} = (1.0025)^{120} \approx 1.348850
\]
Now we can substitute this into the formula:
\[
S = 95 \left( \frac{1.348850 - 1}{0.0025} \right)
\]
Calculating the numerator:
\[
1.348850 - 1 = 0.348850
\]
Now we will substitute this back into the equation:
\[
S = 95 \left( \frac{0.348850}{0.0025} \right) = 95 \times 139.54 \approx 13205.36
\]
Finally, let's get the total amount:
\[
S \approx 13205.36
\]
When we round to two decimal places, we find:
\[
S \approx 13205.36
\]
So, looking at the closest answer choices provided, we have \( S \approx 13275.43 \).
Thus, the final answer is:
**$13,275.43**