객관식
What is the present value of a \$750 payment to be received in three years if the discount rate is 5\% per year, compounded annually?
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10. Time Value of Money
Time Value of Money Equations
객관식
Approximately what annual interest rate, compounded annually, is needed to double an investment over eight years?
A
12%
B
6.5%
C
8.7%
D
9%
검증된 단계별 안내1
Step 1: Understand the problem. The goal is to determine the annual interest rate required to double an investment over eight years using the formula for compound interest. The formula is: \( A = P(1 + r)^t \), where \( A \) is the future value, \( P \) is the principal amount, \( r \) is the annual interest rate, and \( t \) is the time in years.
Step 2: Set up the equation based on the problem. Since the investment needs to double, \( A = 2P \). Substitute \( A = 2P \), \( t = 8 \), and \( P \) into the formula: \( 2P = P(1 + r)^8 \).
Step 3: Simplify the equation. Divide both sides of the equation by \( P \) (assuming \( P \neq 0 \)): \( 2 = (1 + r)^8 \).
Step 4: Solve for \( r \). Take the eighth root of both sides to isolate \( 1 + r \): \( 1 + r = \sqrt[8]{2} \). Then subtract 1 from both sides: \( r = \sqrt[8]{2} - 1 \).
Step 5: Interpret the result. The value of \( r \) represents the annual interest rate needed to double the investment over eight years. Use a calculator or logarithmic methods to approximate \( \sqrt[8]{2} \) if needed.
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