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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