Question
Tyler wants to have 250,000
in his retirement account by the time he retires in 30 years. The interest rate is 5%
. Use the annuity formula to calculate the amount Tyler needs to deposit on a monthly basis in order to reach his goal. When solving, round numbers to the nearest hundred-thousandth. Round your final answer to the nearest cent.
1,666.67
250.50
565.18
300.16
in his retirement account by the time he retires in 30 years. The interest rate is 5%
. Use the annuity formula to calculate the amount Tyler needs to deposit on a monthly basis in order to reach his goal. When solving, round numbers to the nearest hundred-thousandth. Round your final answer to the nearest cent.
1,666.67
250.50
565.18
300.16
Answers
GPT-4o mini
To find out how much Tyler needs to deposit monthly in order to reach his retirement goal of $250,000 in 30 years with an interest rate of 5%, we can use the future value of an ordinary annuity formula:
\[
FV = P \times \left( \frac{(1 + r)^n - 1}{r} \right)
\]
Where:
- \( FV \) = future value of the annuity
- \( P \) = payment amount per period (monthly deposit)
- \( r \) = interest rate per period
- \( n \) = total number of payments
**Given:**
- \( FV = 250,000 \)
- Annual interest rate = 5% → Monthly interest rate (\( r \)) = \( \frac{5\%}{12} = \frac{0.05}{12} \approx 0.0041667 \)
- Time = 30 years → Total number of deposits (\( n \)) = \( 30 \times 12 = 360 \)
Now, we need to rearrange the annuity formula to solve for \( P \):
\[
P = \frac{FV}{\left( \frac{(1 + r)^n - 1}{r} \right)}
\]
Substituting the known values into the formula:
1. Calculate \( (1 + r)^n \):
\[
(1 + 0.0041667)^{360} \approx (1.0041667)^{360} \approx 4.467744
\]
2. Now compute \( (1 + r)^n - 1 \):
\[
4.467744 - 1 \approx 3.467744
\]
3. Now divide by \( r \):
\[
\frac{3.467744}{0.0041667} \approx 831.24
\]
4. Finally, calculate \( P \):
\[
P = \frac{250,000}{831.24} \approx 300.16
\]
Thus, the amount Tyler needs to deposit monthly is approximately **$300.16**, which is already rounded to the nearest cent.
So the final answer is:
**300.16**
\[
FV = P \times \left( \frac{(1 + r)^n - 1}{r} \right)
\]
Where:
- \( FV \) = future value of the annuity
- \( P \) = payment amount per period (monthly deposit)
- \( r \) = interest rate per period
- \( n \) = total number of payments
**Given:**
- \( FV = 250,000 \)
- Annual interest rate = 5% → Monthly interest rate (\( r \)) = \( \frac{5\%}{12} = \frac{0.05}{12} \approx 0.0041667 \)
- Time = 30 years → Total number of deposits (\( n \)) = \( 30 \times 12 = 360 \)
Now, we need to rearrange the annuity formula to solve for \( P \):
\[
P = \frac{FV}{\left( \frac{(1 + r)^n - 1}{r} \right)}
\]
Substituting the known values into the formula:
1. Calculate \( (1 + r)^n \):
\[
(1 + 0.0041667)^{360} \approx (1.0041667)^{360} \approx 4.467744
\]
2. Now compute \( (1 + r)^n - 1 \):
\[
4.467744 - 1 \approx 3.467744
\]
3. Now divide by \( r \):
\[
\frac{3.467744}{0.0041667} \approx 831.24
\]
4. Finally, calculate \( P \):
\[
P = \frac{250,000}{831.24} \approx 300.16
\]
Thus, the amount Tyler needs to deposit monthly is approximately **$300.16**, which is already rounded to the nearest cent.
So the final answer is:
**300.16**