Find Initial Deposit for $50,000 Future Value with 7.5% Quarterly Compounding Over 8 Years

Question

Find how much money should be deposited in a bank paying interest at the rate of 7.5%/year compounded quarterly so that at the end of 8 years, the accumulated amount will be $50,000. (Round your answer to the nearest cent.)

Accepted Answer

$27893.11

Answer

Step 1: Identify the compound interest present value formula: P = A / (1 + r/n)^(nt), where P is the principal (initial deposit), A is the accumulated amount, r is the annual interest rate (decimal), n is the number of compounding periods per year, and t is time in years. Step 2: Convert the annual rate to decimal: r = 7.5% = 0.075. Define other variables: A = $50,000, n = 4 (quarterly compounding), t = 8 years. Step 3: Calculate the compounding factor: (1 + 0.075/4)^(4×8) = (1.01875)^32 ≈ 1.79259. Step 4: Solve for principal: P = $50,000 / 1.79259 ≈ $27893.11 when rounded to the nearest cent.