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 $$ X $$.

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

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