Question 6 of 15, Step 1 of 1 Hale College predicts that in 8 years it will take $$ \$ 200,000 $$ to attend the college for four years. Alexis has a substantial amount of cash and wishes to invest a lump sum of money for her child's college fund. How much should Alexis put aside in an account with an APR of $$ 11 \% $$ compounded monthly in order to have $$ \$ 200,000 $$ in the account in 8 years? Round your answer to the nearest cent, if necessary. Formulas Answer How to enter your answer (opens in new window)

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To determine how much Alexis should invest now to have $$ \$ 200,000 $$ in 8 years with an APR of $$ 11\% $$ compounded monthly, we can use the formula for compound interest:

$$
A = P \left(1 + \frac{r}{n}\right)^{nt}
$$

Where:
- $$ A $$ is the amount of money accumulated after n years, including interest.
- $$ P $$ is the principal amount (the initial amount of money).
- $$ r $$ is the annual interest rate (decimal).
- $$ n $$ is the number of times that interest is compounded per year.
- $$ t $$ is the number of years the money is invested or borrowed.

In this case:
- $$ A = 200,000 $$
- $$ r = 0.11 $$
- $$ n = 12 $$ (since the interest is compounded monthly)
- $$ t = 8 $$

We need to solve for $$ P $$. Rearranging the formula gives us:

$$
P = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}}
$$

Now, we can substitute the known values into the formula to find $$ P $$:

$$
P = \frac{200,000}{\left(1 + \frac{0.11}{12}\right)^{12 \times 8}}
$$

Let's calculate this step by step.

1. Calculate $$ \frac{r}{n} $$.
2. Calculate $$ nt $$.
3. Calculate $$ \left(1 + \frac{r}{n}\right)^{nt} $$.
4. Finally, calculate $$ P $$.

I'll perform these calculations now.
Calculate the value by following steps:
- step0: Calculate:
  $$12\times 8$$
- step1: Multiply the numbers:
  $$96$$
Calculate or simplify the expression $$ 0.11/12 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{0.11}{12}$$
- step1: Convert the expressions:
  $$\frac{\frac{11}{100}}{12}$$
- step2: Multiply by the reciprocal:
  $$\frac{11}{100}\times \frac{1}{12}$$
- step3: Multiply the fractions:
  $$\frac{11}{100\times 12}$$
- step4: Multiply:
  $$\frac{11}{1200}$$
Calculate or simplify the expression $$ (1 + 0.009166666666666667)^(96) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(1+\frac{9166666666666667}{1000000000000000000}\right)^{96}$$
- step1: Add the numbers:
  $$\left(\frac{1009166666666666667}{1000000000000000000}\right)^{96}$$
- step2: Use the properties of exponents:
  $$\frac{1009166666666666667^{96}}{1000000000000000000^{96}}$$
Calculate or simplify the expression $$ 200000/(2.401254) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{200000}{2.401254}$$
- step1: Convert the expressions:
  $$\frac{200000}{\frac{1200627}{500000}}$$
- step2: Multiply by the reciprocal:
  $$200000\times \frac{500000}{1200627}$$
- step3: Multiply:
  $$\frac{200000\times 500000}{1200627}$$
- step4: Multiply:
  $$\frac{100000000000}{1200627}$$
Let's summarize the calculations step by step:

1. **Calculate $$ \frac{r}{n} $$**:
   $$
   \frac{0.11}{12} \approx 0.00916667
   $$

2. **Calculate $$ nt $$**:
   $$
   12 \times 8 = 96
   $$

3. **Calculate $$ \left(1 + \frac{r}{n}\right)^{nt} $$**:
   $$
   \left(1 + 0.00916667\right)^{96} \approx 2.401254
   $$

4. **Calculate $$ P $$**:
   $$
   P = \frac{200,000}{2.401254} \approx 83,289.81
   $$

Thus, Alexis should invest approximately **\$83,289.81** to have \$200,000 in 8 years at an APR of 11% compounded monthly.
Alexis should invest approximately \$83,289.81 now to have \$200,000 in 8 years at an APR of 11% compounded monthly.

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