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
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
Answers
GPT-4o mini
To calculate the future value of an annuity where regular payments are made into an account earning interest, we can use the formula for the future value of an annuity:
\[
S = P \left( \frac{(1 + r/n)^{nt} - 1}{r/n} \right)
\]
Where:
- \( S \) = future value of the annuity
- \( P \) = monthly payment (annuity payment)
- \( r \) = annual interest rate (as a decimal)
- \( n \) = number of times interest is compounded per year
- \( t \) = number of years the money is invested
Given that:
- \( P = 95 \) (monthly payment)
- \( r = 0.03 \) (annual interest rate)
- \( n = 12 \) (interest compounded monthly)
- \( t = 10 \) (years)
We will start by calculating the components of the formula:
1. Calculate \( r/n \):
\[
r/n = 0.03/12 = 0.0025
\]
2. Calculate \( nt \):
\[
nt = 12 \times 10 = 120
\]
3. Now substitute these values into the formula:
\[
S = 95 \left( \frac{(1 + 0.0025)^{120} - 1}{0.0025} \right)
\]
4. Calculate \( (1 + 0.0025)^{120} \):
\[
(1 + 0.0025)^{120} = (1.0025)^{120} \approx 1.34885
\]
5. Calculate the entire fraction:
\[
(1.34885 - 1) = 0.34885
\]
\[
\frac{0.34885}{0.0025} \approx 139.54
\]
6. Finally, calculate \( S \):
\[
S = 95 \times 139.54 \approx 13,257.30
\]
Rounding this value gives approximately \$13,275.43, which matches the second option from your list:
**Response: $13,275.43**
\[
S = P \left( \frac{(1 + r/n)^{nt} - 1}{r/n} \right)
\]
Where:
- \( S \) = future value of the annuity
- \( P \) = monthly payment (annuity payment)
- \( r \) = annual interest rate (as a decimal)
- \( n \) = number of times interest is compounded per year
- \( t \) = number of years the money is invested
Given that:
- \( P = 95 \) (monthly payment)
- \( r = 0.03 \) (annual interest rate)
- \( n = 12 \) (interest compounded monthly)
- \( t = 10 \) (years)
We will start by calculating the components of the formula:
1. Calculate \( r/n \):
\[
r/n = 0.03/12 = 0.0025
\]
2. Calculate \( nt \):
\[
nt = 12 \times 10 = 120
\]
3. Now substitute these values into the formula:
\[
S = 95 \left( \frac{(1 + 0.0025)^{120} - 1}{0.0025} \right)
\]
4. Calculate \( (1 + 0.0025)^{120} \):
\[
(1 + 0.0025)^{120} = (1.0025)^{120} \approx 1.34885
\]
5. Calculate the entire fraction:
\[
(1.34885 - 1) = 0.34885
\]
\[
\frac{0.34885}{0.0025} \approx 139.54
\]
6. Finally, calculate \( S \):
\[
S = 95 \times 139.54 \approx 13,257.30
\]
Rounding this value gives approximately \$13,275.43, which matches the second option from your list:
**Response: $13,275.43**