Supposed that your paychecks are issued on the 1st and the 15th of each month, and that you deposit a portion of each paycheck into an account that earns 8% interest. How much would you need to deposit from each paycheck in order to have $50,000 in the account in 6 years?$
Question
Supposed that your paychecks are issued on the 1st and the 15th of each month, and that you deposit a portion of each paycheck into an account that earns 8% interest. How much would you need to deposit from each paycheck in order to have
Solution
To solve this problem, we need to use the formula for the future value of an annuity due, which is a series of equal payments made at the beginning of each period. The formula is:
FV = P * [(1 + r/n)^(nt) - 1] / (r/n) * (1 + r/n)
where: FV = future value of the annuity P = amount deposited each period r = annual interest rate n = number of compounding periods per year t = number of years
We know that FV = $50,000, r = 8% or 0.08, n = 24 (since there are 24 pay periods in a year), and t = 6. We want to solve for P.
First, let's simplify the equation:
FV = P * [(1 + 0.08/24)^(24*6) - 1] / (0.08/24) * (1 + 0.08/24)
Next, calculate the values in the brackets:
FV = P * [2.59374 - 1] / 0.003333 * 1.003333
Simplify further:
FV = P * 1.59374 / 0.003333 * 1.003333
Now, solve for P:
P = FV / (1.59374 / 0.003333 * 1.003333)
P = $50,000 / (1.59374 / 0.003333 * 1.003333)
P = $50,000 / 5.318
P = $9,400.15
So, you would need to deposit approximately 50,000 in the account in 6 years.
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