Question
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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 \( X \).

Ask by Craig Ramos. in the United States
Mar 12,2025

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Answer

Account \( Y \) would earn Monica about \$473.62 more interest than Account \( X \) after 7 years.

Solution

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 \( X \).

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Bonus Knowledge

To calculate the interest earned from both accounts, we can use the formulas for simple and compound interest. For Account \(X\) with simple interest: Interest \(I_X = P \times r \times t\) = \$1752 × 0.078 × 7 = \$960.96. For Account \(Y\) with compound interest: Amount \(A_Y = P(1 + r)^t\) = \$1752 × (1 + 0.089)^7 = \$1752 × (1.089)^7 ≈ \$1752 × 1.7461 ≈ \$3,058.23. Interest \(I_Y = A_Y - P\) = \$3,058.23 - \$1752 = \$1,306.23. Finally, calculating the difference: \$1,306.23 (Account Y) - \$960.96 (Account X) ≈ \$345.27. So, Account \(Y\) would earn Monica about \(\$ \square\) \(345.27\) more interest than Account \(X\). Can you believe how much compounding can work to your advantage? It’s like planting a tree; the more you nurture it, the bigger it grows! Speaking of gardening, did you know that understanding the power of compounding can help you make informed choices not just in finance but also in life? Just like the care you put into your finances, nurturing personal growth can lead to thriving results!

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