After getting her first job, a college graduate wants to begin saving money so she can pay cash for a new car. She wants to save enough each month to have $21,000 at the end of 4 years, and her savings account pays 4% interest, compounded monthly. How much will she have to save each month to reach her goal? Express your answer in $ to the nearest whole $.
To find out how much the college graduate needs to save each month, we can use the formula for future value of an ordinary annuity:
FV = P * (((1 + r)^n) - 1) / r
Where:
FV = future value (amount she wants to have at the end of 4 years)
P = monthly saving amount
r = monthly interest rate (4% divided by 12 to get monthly rate)
n = number of months (4 years multiplied by 12 months per year)
Let's plug in the values and solve for P:
21,000 = P * (((1 + 0.04/12)^(4 * 12)) - 1) / (0.04/12)
First, simplify the expression inside the parentheses:
21,000 = P * ((1.003333)^48 - 1) / (0.003333)
Find the value of (1.003333)^48:
(1.003333)^48 ≈ 1.181137
Now substitute this value back into the equation:
21,000 = P * (1.181137 - 1) / (0.003333)
Simplify the expression inside the parentheses:
21,000 = P * 0.181137 / 0.003333
Divide both sides by 0.181137 / 0.003333 to solve for P:
P = 21,000 / (0.181137 / 0.003333)
P ≈ 185.71
Therefore, the college graduate needs to save approximately $185.71 each month to have $21,000 at the end of 4 years.