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\%

Answer: (A)

Verified step by step guidance

1. 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.
2. 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.
3. 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: \( r = (\frac{A}{P})^{\frac{1}{n}} - 1 \).
4. Step 4: Substitute the known values into the rearranged formula. Replace \( A \) with 7260, \( P \) with 6000, and \( n \) with 2: \( r = (\frac{7260}{6000})^{\frac{1}{2}} - 1 \).
5. 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.