Your 31-year-old brother tells you he wants to have $1,733,597 in his retirement fund when he retires at age 68. How much should you tell him to save per month if he can invest at 9.6% APR compounded monthly?
Group of answer choices

$415.31

$394.55

$419.47

$436.08

To solve this, we can use the future value of an ordinary annuity formula, which calculates the amount of savings needed to reach a certain future value, given a regular monthly contribution and an interest rate. The formula is:

\[
FV = P \times \frac{(1 + r)^n - 1}{r}
\]

Where:
- $FV$ is the future value of the retirement fund, which is $1,733,597.
- $P$ is the monthly contribution (what we are trying to find).
- $r$ is the monthly interest rate.
- $n$ is the number of months the contributions are made.

### Step 1: Calculate the variables
- **APR (Annual Percentage Rate) = 9.6%**, so the monthly interest rate is:
  \[
  r = \frac{9.6\%}{12} = \frac{0.096}{12} = 0.008
  \]
- **Number of months**: He will be saving from age 31 to 68, which is 37 years, so:
  \[
  n = 37 \times 12 = 444 \text{ months}
  \]
- The future value $FV$ is $1,733,597.

### Step 2: Solve for $P$ (the monthly contribution)
Rearrange the formula to solve for $P$:

\[
P = \frac{FV \times r}{(1 + r)^n - 1}
\]

Substitute the values:

\[
P = \frac{1,733,597 \times 0.008}{(1 + 0.008)^{444} - 1}
\]

Now, we can calculate this step by step.

1. Calculate $(1 + 0.008)^{444}$:
   \[
   (1.008)^{444} = 21.581
   \]
2. Subtract 1:
   \[
   21.581 - 1 = 20.581
   \]
3. Multiply the future value by the monthly interest rate:
   \[
   1,733,597 \times 0.008 = 13,868.776
   \]
4. Finally, divide:
   \[
   P = \frac{13,868.776}{20.581} = 673.91
   \]

It seems there was an error earlier—likely due to misinterpretation of the annuity equation. Let me

Your 31-year-old brother tells you he wants to have $1,733,597 in his retirement fund when he retires at age 68. How much should you tell him to save per month if he can invest at 9.6% APR compounded monthly? Group of answer choices: $415.31, $394.55, $419.47, $436.08

Learn how to calculate the monthly savings needed to reach a retirement goal of $1,733,597 using compound interest at a 9.6% APR, compounded monthly. This finance problem applies annuity formulas and compound interest concepts, suitable for advanced high school or college-level students.

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