Question
LaKeisha wants to save $500,000 for her retirement and plans to make monthly deposits into an annuity for the next 30 years. If the annuity interest rate is 4 percent, calculate how much LaKeisha should invest every month to reach her goal. Round the answer to the nearest whole number.(1 point)
Responses
$3,678
$3,678
$1,583
$1,583
$1,876
$1,876
$2,387
Responses
$3,678
$3,678
$1,583
$1,583
$1,876
$1,876
$2,387
Answers
GPT-4o mini
To calculate how much LaKeisha needs to invest every month in an annuity to reach her retirement goal of $500,000 over 30 years with an interest rate of 4%, we can use the future value of an annuity formula:
\[
FV = P \times \frac{(1 + r)^n - 1}{r}
\]
Where:
- \( FV \) is the future value of the annuity ($500,000)
- \( P \) is the monthly deposit (the value we're trying to find)
- \( r \) is the monthly interest rate (annual rate divided by 12)
- \( n \) is the total number of deposits (number of years multiplied by 12)
### Step 1: Calculate the monthly interest rate and total number of deposits
\[
\text{Annual interest rate} = 4\% = 0.04
\]
\[
r = \frac{0.04}{12} \approx 0.003333
\]
\[
n = 30 \times 12 = 360 \text{ months}
\]
### Step 2: Rearrange the formula to solve for \( P \)
\[
500,000 = P \times \frac{(1 + 0.003333)^{360} - 1}{0.003333}
\]
### Step 3: Calculate \((1 + r)^n\)
\[
(1 + 0.003333)^{360} \approx 3.243
\]
### Step 4: Plug this value back into the formula
\[
500,000 = P \times \frac{3.243 - 1}{0.003333}
\]
\[
500,000 = P \times \frac{2.243}{0.003333} \approx P \times 672.90
\]
### Step 5: Solve for \( P \)
\[
P = \frac{500,000}{672.90} \approx 743.8
\]
Thus, the amount LaKeisha needs to save every month is approximately **$743.8**.
### Rechecking the options provided:
None of the provided options ($3,678, $1,583, $1,876, $2,387) seem to match.
Re-evaluating utilizing a financial calculator or software could provide a more exact figure.
Given this task, it appears I made an assumption based on the expected answers. Let's calculate \( P \) precisely.
\[
500,000 = P \times 672.9 \rightarrow P = \frac{500,000}{672.9}
\]
Calculating,
\[
P \approx 743.8 (Rounding does not find an option)
\]
None of the provided responses directly correspond to the calculated saving required for LaKeisha. The calculation indicates she needs considerably less than mentioned in the options. The values shown in the options may pertain to a different specific scenario or require a review since they do not align with the doable calculations under the given constraints.
Confirming once more from varied equations or definitions on annuities could clarify misaligning expectations between calculated annuity values versus options available in this task.
Please review the statement or parameters, or workout scenarios for an accurate endorsement of LaKeisha's needed investment over 30 years against the stipulated 4% annuity. Always available to assist recalibrating aspects toward correct values as revisited through various percentage perspectives or savings method enactments!
\[
FV = P \times \frac{(1 + r)^n - 1}{r}
\]
Where:
- \( FV \) is the future value of the annuity ($500,000)
- \( P \) is the monthly deposit (the value we're trying to find)
- \( r \) is the monthly interest rate (annual rate divided by 12)
- \( n \) is the total number of deposits (number of years multiplied by 12)
### Step 1: Calculate the monthly interest rate and total number of deposits
\[
\text{Annual interest rate} = 4\% = 0.04
\]
\[
r = \frac{0.04}{12} \approx 0.003333
\]
\[
n = 30 \times 12 = 360 \text{ months}
\]
### Step 2: Rearrange the formula to solve for \( P \)
\[
500,000 = P \times \frac{(1 + 0.003333)^{360} - 1}{0.003333}
\]
### Step 3: Calculate \((1 + r)^n\)
\[
(1 + 0.003333)^{360} \approx 3.243
\]
### Step 4: Plug this value back into the formula
\[
500,000 = P \times \frac{3.243 - 1}{0.003333}
\]
\[
500,000 = P \times \frac{2.243}{0.003333} \approx P \times 672.90
\]
### Step 5: Solve for \( P \)
\[
P = \frac{500,000}{672.90} \approx 743.8
\]
Thus, the amount LaKeisha needs to save every month is approximately **$743.8**.
### Rechecking the options provided:
None of the provided options ($3,678, $1,583, $1,876, $2,387) seem to match.
Re-evaluating utilizing a financial calculator or software could provide a more exact figure.
Given this task, it appears I made an assumption based on the expected answers. Let's calculate \( P \) precisely.
\[
500,000 = P \times 672.9 \rightarrow P = \frac{500,000}{672.9}
\]
Calculating,
\[
P \approx 743.8 (Rounding does not find an option)
\]
None of the provided responses directly correspond to the calculated saving required for LaKeisha. The calculation indicates she needs considerably less than mentioned in the options. The values shown in the options may pertain to a different specific scenario or require a review since they do not align with the doable calculations under the given constraints.
Confirming once more from varied equations or definitions on annuities could clarify misaligning expectations between calculated annuity values versus options available in this task.
Please review the statement or parameters, or workout scenarios for an accurate endorsement of LaKeisha's needed investment over 30 years against the stipulated 4% annuity. Always available to assist recalibrating aspects toward correct values as revisited through various percentage perspectives or savings method enactments!