18. Dakota earned \( \$ 15.75 \) in interest in Account A and \( \$ 28.00 \) in interest in Account B after 21 months. If the simple interest rate is for \( 3.0 \% \) for Account A and \( 4.0 \% \) for Account B, which account had the greater principal? Explain.

To determine which account had the greater principal, we can use the simple interest formula: \[ I = P \times r \times t \] where \( I \) is the interest earned, \( P \) is the principal amount, \( r \) is the annual interest rate (in decimal), and \( t \) is the time in years. For Account A, with an interest of \( \$ 15.75 \) at \( 3.0\% \) for \( 21 \) months (or \( 1.75 \) years), the calculation for the principal \( P_A \) is: \[ 15.75 = P_A \times 0.03 \times 1.75 \] Solving this, we find \( P_A \approx 300.00 \). For Account B, with an interest of \( \$ 28.00 \) at \( 4.0\% \) for \( 21 \) months, the calculation for the principal \( P_B \) is: \[ 28.00 = P_B \times 0.04 \times 1.75 \] Solving this, we find \( P_B \approx 200.00 \). Therefore, Account A had the greater principal. Knowing how interest rates work in practice can significantly inform your financial decisions. For instance, understanding that higher interest rates don’t always mean you earn more interest can help you make smarter savings decisions. Always compare the total interest earned across different accounts, considering both the rate and the principal, to understand which is more beneficial for your savings. When calculating principal amounts using the interest formula, a common mistake is forgetting to convert the interest rate into decimal form. Always remember that \( 3.0\% \) is expressed as \( 0.03 \) and \( 4.0\% \) as \( 0.04 \)—negligent calculations can lead to incorrect conclusions about potential earnings! Additionally, ensure that your time period is consistently measured in years, turning months into fractions of the year when necessary.