Multiple Choice
If you invest \$823 at an annual interest rate of 3\% compounded annually for 5 years, what will be the future value of your investment? (Use the formula: \( FV = PV \times (1 + r)^n \))
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10. Time Value of Money
Time Value of Money Equations
Multiple Choice
If Tom borrows \$6,000 at a compound interest rate and must repay \$7,260 after 2 years, what is the annual compound interest rate?
A
12\%
B
15\%
C
8\%
D
10\%
Verified step by step guidance1
Step 1: Understand the formula for compound interest. The formula is: \( A = P(1 + r)^n \), where \( A \) is the amount after interest, \( P \) is the principal amount, \( r \) is the annual interest rate (in decimal form), and \( n \) is the number of years.
Step 2: Identify the values given in the problem. Here, \( A = 7260 \), \( P = 6000 \), and \( n = 2 \). The goal is to solve for \( r \), the annual compound interest rate.
Step 3: Rearrange the formula to isolate \( r \). Divide both sides of the equation by \( P \): \( \frac{A}{P} = (1 + r)^n \). Then take the \( n \)-th root of both sides: \( (\frac{A}{P})^{\frac{1}{n}} = 1 + r \). Finally, subtract 1 from both sides: \( r = (\frac{A}{P})^{\frac{1}{n}} - 1 \).
Step 4: Substitute the known values into the rearranged formula. Replace \( A \) with 7260, \( P \) with 6000, and \( n \) with 2: \( r = (\frac{7260}{6000})^{\frac{1}{2}} - 1 \).
Step 5: Simplify the expression step by step. First, calculate \( \frac{7260}{6000} \), then take the square root (\( \frac{1}{2} \)-th power), and finally subtract 1 to find \( r \). Convert \( r \) to a percentage by multiplying by 100.
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