Monica wants to open a savings account with a deposit of \$1752. Monica will not make any additional deposits or withdrawals after she opens the account. Her bank offers two different savings accounts. - Account \( X \) pays \( 7.8 \% \) simple annual interest. - Account \( Y \) pays 8.9 interest compounded annually. Complete the statement about these two accounts after 7 years. Account \( Y \) would earn Monica about \( \$ \) \( \square \) 357.87 more interest than Account \( X \).

To determine how much more interest Account \( Y \) would earn compared to Account \( X \) after 7 years, we will calculate the interest earned from both accounts using the respective interest formulas. 1. **Calculate the interest for Account \( X \)** (simple interest): The formula for simple interest is: \[ I_X = P \cdot r \cdot t \] where: - \( I_X \) = interest earned from Account \( X \) - \( P = 1752 \) (the principal amount) - \( r = 0.078 \) (the annual interest rate for Account \( X \)) - \( t = 7 \) (the time in years) 2. **Calculate the interest for Account \( Y \)** (compound interest): The formula for compound interest is: \[ A_Y = P \cdot (1 + r)^t \] where: - \( A_Y \) = total amount in Account \( Y \) after \( t \) years - \( r = 0.089 \) (the annual interest rate for Account \( Y \)) - \( t = 7 \) (the time in years) - The interest earned from Account \( Y \) can be calculated as: \[ I_Y = A_Y - P \] Now, let's perform the calculations for both accounts. ### Step 1: Calculate Interest for Account \( X \) \[ I_X = 1752 \cdot 0.078 \cdot 7 \] ### Step 2: Calculate Total Amount for Account \( Y \) \[ A_Y = 1752 \cdot (1 + 0.089)^7 \] \[ I_Y = A_Y - 1752 \] ### Step 3: Calculate the Difference in Interest \[ \text{Difference} = I_Y - I_X \] Now, let's compute these values. Calculate the value by following steps: - step0: Calculate: \(1752\times 0.078\times 7\) - step1: Multiply the terms: \(136.656\times 7\) - step2: Multiply the numbers: \(956.592\) Calculate or simplify the expression \( 1752 * (1 + 0.089)^7 \). Calculate the value by following steps: - step0: Calculate: \(1752\left(1+0.089\right)^{7}\) - step1: Add the numbers: \(1752\times 1.089^{7}\) - step2: Convert the expressions: \(1752\left(\frac{1089}{1000}\right)^{7}\) - step3: Simplify: \(1752\times \frac{1089^{7}}{1000^{7}}\) - step4: Rewrite the expression: \(8\times 219\times \frac{1089^{7}}{1000^{7}}\) - step5: Rewrite the expression: \(8\times 219\times \frac{1089^{7}}{8^{7}\times 125^{7}}\) - step6: Reduce the numbers: \(219\times \frac{1089^{7}}{8^{6}\times 125^{7}}\) - step7: Multiply the fractions: \(\frac{219\times 1089^{7}}{8^{6}\times 125^{7}}\) Calculate or simplify the expression \( 3182.213106 - 1752 \). Calculate the value by following steps: - step0: Calculate: \(3182.213106-1752\) - step1: Subtract the numbers: \(1430.213106\) Now that we have the necessary calculations, let's summarize the results: 1. **Interest earned from Account \( X \)**: \[ I_X = 956.59 \] 2. **Total amount in Account \( Y \)** after 7 years: \[ A_Y \approx 3182.21 \] Therefore, the interest earned from Account \( Y \) is: \[ I_Y = A_Y - 1752 \approx 3182.21 - 1752 = 1430.21 \] 3. **Difference in interest between Account \( Y \) and Account \( X \)**: \[ \text{Difference} = I_Y - I_X \approx 1430.21 - 956.59 = 473.62 \] Thus, after 7 years, Account \( Y \) would earn Monica about \( \$ \) \( 473.62 \) more interest than Account \( X \). ### Final Statement: Account \( Y \) would earn Monica about \( \$ \) \( 473.62 \) more interest than Account \( X \).