To determine which type of interest, simple or compound, yields a higher balance after 6 years for an investment of $17,000 at an interest rate of 3%, we'll calculate both.
Simple Interest Calculation:
The formula for simple interest is: \[ A = P(1 + rt) \] where:
- \( A \) = total amount after time \( t \)
- \( P \) = principal amount (initial investment)
- \( r \) = annual interest rate (as a decimal)
- \( t \) = time (in years)
For our scenario:
- \( P = 17,000 \)
- \( r = 0.03 \)
- \( t = 6 \)
Plugging in these values: \[ A = 17,000(1 + 0.03 \times 6) \] \[ A = 17,000(1 + 0.18) \] \[ A = 17,000 \times 1.18 \] \[ A = 20,066 \]
Compound Interest Calculation:
The formula for compound interest is: \[ A = P(1 + r)^t \]
Using the same \( P \), \( r \), and \( t \): \[ A = 17,000(1 + 0.03)^6 \] \[ A = 17,000(1.03)^6 \]
Calculating \( (1.03)^6 \): \[ (1.03)^6 \approx 1.194052 \] Now, substitute back: \[ A \approx 17,000 \times 1.194052 \] \[ A \approx 20,298.89 \]
Summary:
- Simple Interest Balance after 6 years: \( \approx 20,066 \)
- Compound Interest Balance after 6 years: \( \approx 20,298.89 \)
Thus, compound interest at 3% is the better interest earning type, with a resulting balance of approximately $20,298.89 after 6 years of investment.