Hale College predicts that in 1818 years it will take $200,000$⁢200,000 to attend the college for four years. Hannah has a substantial amount of cash and wishes to invest a lump sum of money for her child’s college fund. How much should Hannah put aside in an account with an APR of 12%12% compounded monthly in order to have $200,000$⁢200,000 in the account in 1818 years? Round your answer to the nearest cent, if necessary.

To solve this problem, we need to use the formula for compound interest, which is:

A = P(1 + r/n)^(nt)

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

We know that A = $200,000, r = 12% or 0.12 (in decimal), n = 12 (compounded monthly), and t = 18 years. We need to find P.

Rearranging the formula to solve for P gives us:

P = A / (1 + r/n)^(nt)

Substituting the known values gives us:

P = 200,000 / (1 + 0.12/12)^(12*18)

Calculating the right side of the equation gives us the amount Hannah needs to invest.

P = 200,000 / (1 + 0.01)^(216) P = 200,000 / (1.01)^216 P = 200,000 / 14.97 P = $13,363.40

So, Hannah needs to invest approximately $13,363.40 in the account.

To solve this problem, we need to use the formula for compound interest, which is:

A = P(1 + r/n)^(nt)

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

We know that A = $200,000, r = 12% or 0.12 (in decimal), n = 12 (since the interest is compounded monthly), and t = 18 years. We need to find P.

So, we rearrange the formula to solve for P:

P = A / (1 + r/n)^(nt)

Substituting the given values:

P = 200,000 / (1 + 0.12/12)^(12*18)

Now, calculate the expression in the parentheses:

= 200,000 / (1 + 0.01)^(216)

= 200,000 / (1.01)^216

Now, calculate the exponent:

= 200,000 / 14.97

Finally, divide 200,000 by 14.97 to find P:

P = $13,361.39

So, Hannah should put aside approximately $13,361.39 in the account.