Question 2 John takes out a loan for $$ \$ 9300 $$ at $$ 9 \% $$ interest compounded monthly and is making payments of $$ \$ 88 $$ a month. Calculate his remaining balance after 18 months. John's balance due after 18 months will be $$ \$ $$ Time Value of Money Solver Enter the given values. $$ \mathrm{N}:=00 $$

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1. Define the variables:  
   - Principal: $$ P = 9300 $$  
   - Annual interest rate: $$ 9\% $$  
   - Monthly interest rate:  
     $$
     i = \frac{9\%}{12} = \frac{0.09}{12} = 0.0075
     $$
   - Monthly payment: $$ \text{PMT} = 88 $$  
   - Number of months: $$ n = 18 $$

2. The balance remaining on a loan after $$ n $$ payments (with payments made at the end of each month) is given by the formula:  
   $$
   B = P(1+i)^n - \text{PMT} \cdot \frac{(1+i)^n - 1}{i}
   $$

3. Substitute the given values into the formula:  
   $$
   B = 9300(1.0075)^{18} - 88 \cdot \frac{(1.0075)^{18} - 1}{0.0075}
   $$

4. Calculate the compound factor:  
   $$
   (1.0075)^{18} \approx 1.1441
   $$

5. Calculate the first term (the compounded principal):  
   $$
   9300(1.1441) \approx 9300 \times 1.1441 \approx 10641.3
   $$

6. Calculate the second term (the effect of the payments):  
   - First compute the numerator:  
     $$
     (1.0075)^{18} - 1 \approx 1.1441 - 1 = 0.1441
     $$
   - Then compute the division:  
     $$
     \frac{0.1441}{0.0075} \approx 19.2133
     $$
   - Multiply by the monthly payment:  
     $$
     88 \times 19.2133 \approx 1690.67
     $$

7. Subtract the total amount repaid from the compounded balance to find the remaining balance:  
   $$
   B \approx 10641.3 - 1690.67 \approx 8950.63
   $$

8. Therefore, John's balance due after 18 months is approximately:  
   $$
   \$8950.63
   $$
John's remaining balance after 18 months is approximately \$8,950.63.

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