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
Answers
GPT-4o mini
To calculate Justin's annuity savings over the course of 25 years with monthly contributions, we can use the formula for the future value of an annuity:
\[
S = P \times \left( \frac{(1 + r/n)^{nt} - 1}{r/n} \right)
\]
Where:
- \(S\) = future value of the annuity
- \(P\) = payment amount per period (monthly contribution)
- \(r\) = annual interest rate (as a decimal)
- \(n\) = number of times interest is compounded per year
- \(t\) = number of years
In this case:
- \(P = 208\) (monthly contribution)
- \(r = 0.05\) (5% annual interest rate)
- \(n = 12\) (interest is compounded monthly)
- \(t = 25\) (25 years)
Now, let's plug in the values into the formula:
\[
S = 208 \times \left( \frac{(1 + \frac{0.05}{12})^{12 \times 25} - 1}{\frac{0.05}{12}} \right)
\]
Calculating step-by-step:
1. Calculate \(r/n\):
\[
r/n = \frac{0.05}{12} \approx 0.0041667
\]
2. Calculate \(nt\):
\[
nt = 12 \times 25 = 300
\]
3. Calculate \((1 + r/n)^{nt}\):
\[
(1 + 0.0041667)^{300} \approx 4.4867
\]
4. Now substitute back into the formula:
\[
S = 208 \times \left( \frac{4.4867 - 1}{0.0041667} \right)
\]
5. Calculate \((4.4867 - 1) / 0.0041667\):
\[
(4.4867 - 1) = 3.4867
\]
\[
\frac{3.4867}{0.0041667} \approx 836.008
\]
6. Finally, multiply by \(P\):
\[
S \approx 208 \times 836.008 \approx 173,054.66
\]
Therefore, the future value of Justin's annuity savings after 25 years would be approximately **$173,054.66**.
None of the provided options exactly match the calculation, suggesting there may have been a misunderstanding or an error in the responses provided.
\[
S = P \times \left( \frac{(1 + r/n)^{nt} - 1}{r/n} \right)
\]
Where:
- \(S\) = future value of the annuity
- \(P\) = payment amount per period (monthly contribution)
- \(r\) = annual interest rate (as a decimal)
- \(n\) = number of times interest is compounded per year
- \(t\) = number of years
In this case:
- \(P = 208\) (monthly contribution)
- \(r = 0.05\) (5% annual interest rate)
- \(n = 12\) (interest is compounded monthly)
- \(t = 25\) (25 years)
Now, let's plug in the values into the formula:
\[
S = 208 \times \left( \frac{(1 + \frac{0.05}{12})^{12 \times 25} - 1}{\frac{0.05}{12}} \right)
\]
Calculating step-by-step:
1. Calculate \(r/n\):
\[
r/n = \frac{0.05}{12} \approx 0.0041667
\]
2. Calculate \(nt\):
\[
nt = 12 \times 25 = 300
\]
3. Calculate \((1 + r/n)^{nt}\):
\[
(1 + 0.0041667)^{300} \approx 4.4867
\]
4. Now substitute back into the formula:
\[
S = 208 \times \left( \frac{4.4867 - 1}{0.0041667} \right)
\]
5. Calculate \((4.4867 - 1) / 0.0041667\):
\[
(4.4867 - 1) = 3.4867
\]
\[
\frac{3.4867}{0.0041667} \approx 836.008
\]
6. Finally, multiply by \(P\):
\[
S \approx 208 \times 836.008 \approx 173,054.66
\]
Therefore, the future value of Justin's annuity savings after 25 years would be approximately **$173,054.66**.
None of the provided options exactly match the calculation, suggesting there may have been a misunderstanding or an error in the responses provided.