Multiple Choice
Raul deposited \$5,000 into a savings account that pays an annual interest rate of 6%, compounded annually. After 3 years, how much interest did Raul earn through compound interest in his savings account?
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10. Time Value of Money
Time Value of Money Equations
Multiple Choice
Approximately what annual interest rate, compounded annually, is needed to double an investment over six years?
A
12%
B
12.25%
C
8.33%
D
10%
Verified step by step guidance1
Step 1: Understand the problem. The goal is to determine the annual interest rate, compounded annually, required to double an investment over six years. This involves using the formula for compound interest and solving for the interest rate.
Step 2: Recall the compound interest formula: \( A = P(1 + r)^t \), where \( A \) is the future value, \( P \) is the principal amount, \( r \) is the annual interest rate (as a decimal), and \( t \) is the time in years. In this case, \( A \) is twice \( P \), and \( t \) is 6 years.
Step 3: Substitute the known values into the formula: \( 2P = P(1 + r)^6 \). Simplify by dividing both sides by \( P \): \( 2 = (1 + r)^6 \). This equation will help us solve for \( r \).
Step 4: To isolate \( r \), take the sixth root (or raise both sides to the power of \( \frac{1}{6} \)): \( (2)^{\frac{1}{6}} = 1 + r \). Then subtract 1 from both sides to find \( r \): \( r = (2)^{\frac{1}{6}} - 1 \).
Step 5: Convert \( r \) into a percentage by multiplying the decimal result by 100. This will give the annual interest rate required to double the investment over six years.
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