Question
Justin contributes $208 each month to a savings account that earns 5% annual interest. Calculate his annuity savings over the course of 25 years if the interest is compounded monthly.
Use S=P((1+rn)nt−1rn)
.
(1 point)
Responses
$9.927.23
$9.927.23
$65,520.00
$65,520.00
$62,660.00
$62,660.00
$123,866.02
Use S=P((1+rn)nt−1rn)
.
(1 point)
Responses
$9.927.23
$9.927.23
$65,520.00
$65,520.00
$62,660.00
$62,660.00
$123,866.02
Answers
GPT-4o mini
To calculate the future value of an annuity with monthly contributions, we use the formula:
\[
S = P \left( \frac{(1 + r/n)^{nt} - 1}{r/n} \right)
\]
Where:
- \( S \) = future value of the annuity
- \( P \) = amount contributed per period
- \( r \) = annual interest rate (as a decimal)
- \( n \) = number of compounding periods per year
- \( t \) = number of years
Given:
- \( P = 208 \)
- \( r = 0.05 \) (5% annual interest)
- \( n = 12 \) (monthly compounding)
- \( t = 25 \)
Now we can plug the values into the formula:
1. Calculate \( r/n\):
\[
r/n = \frac{0.05}{12} = 0.00416667
\]
2. Calculate \( nt \):
\[
nt = 12 \times 25 = 300
\]
3. Calculate \( (1 + r/n)^{nt} \):
\[
(1 + 0.00416667)^{300} \approx 4.391432
\]
4. Now plug everything back into the equation:
\[
S = 208 \left( \frac{4.391432 - 1}{0.00416667} \right)
\]
\[
S = 208 \left( \frac{3.391432}{0.00416667} \right) \approx 208 \times 813.43 \approx 169,169.84
\]
Therefore, the future value of Justin's annuity savings over the course of 25 years, compounded monthly, is approximately **$169,169.84**.
None of the responses you provided match this number; make sure to check your calculation method or the options available.
\[
S = P \left( \frac{(1 + r/n)^{nt} - 1}{r/n} \right)
\]
Where:
- \( S \) = future value of the annuity
- \( P \) = amount contributed per period
- \( r \) = annual interest rate (as a decimal)
- \( n \) = number of compounding periods per year
- \( t \) = number of years
Given:
- \( P = 208 \)
- \( r = 0.05 \) (5% annual interest)
- \( n = 12 \) (monthly compounding)
- \( t = 25 \)
Now we can plug the values into the formula:
1. Calculate \( r/n\):
\[
r/n = \frac{0.05}{12} = 0.00416667
\]
2. Calculate \( nt \):
\[
nt = 12 \times 25 = 300
\]
3. Calculate \( (1 + r/n)^{nt} \):
\[
(1 + 0.00416667)^{300} \approx 4.391432
\]
4. Now plug everything back into the equation:
\[
S = 208 \left( \frac{4.391432 - 1}{0.00416667} \right)
\]
\[
S = 208 \left( \frac{3.391432}{0.00416667} \right) \approx 208 \times 813.43 \approx 169,169.84
\]
Therefore, the future value of Justin's annuity savings over the course of 25 years, compounded monthly, is approximately **$169,169.84**.
None of the responses you provided match this number; make sure to check your calculation method or the options available.