Question 3 John takes out a loan of $$ \$ 11000 $$ that charges $$ 9 \% $$ interest compounded monthly. If John makes $$ \$ 100 $$ monthly payments, determine how long it will take him to pay off the loan. Round your answer up. John will pay off the $$ \$ 11000 $$ loan after Time Value of Money Solver Enter the given values. $$ \mathrm{N}:=\quad 0 $$

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We start with the present value of an annuity formula for a loan:

$$
\text{PV} = \text{PMT} \cdot \frac{1 - (1+i)^{-n}}{i}
$$

Here, 
- $$\text{PV} = \$11000$$,
- $$\text{PMT} = \$100$$,
- The monthly interest rate is $$ i = \frac{0.09}{12} = 0.0075 $$,
- and $$ n $$ is the number of months.

Substitute the values in:

$$
11000 = 100 \cdot \frac{1 - (1.0075)^{-n}}{0.0075}
$$

Multiply both sides by $$0.0075$$:

$$
11000 \times 0.0075 = 100 \left( 1 - (1.0075)^{-n} \right)
$$

$$
82.5 = 100 \left( 1 - (1.0075)^{-n} \right)
$$

Divide both sides by $$100$$:

$$
0.825 = 1 - (1.0075)^{-n}
$$

Solve for $$(1.0075)^{-n}$$:

$$
(1.0075)^{-n} = 1 - 0.825 = 0.175
$$

Take the natural logarithm of both sides:

$$
\ln \left((1.0075)^{-n}\right) = \ln (0.175)
$$

Utilize the power rule for logarithms:

$$
-n \ln(1.0075) = \ln(0.175)
$$

Now, solve for $$ n $$:

$$
n = -\frac{\ln(0.175)}{\ln(1.0075)}
$$

Calculating the logarithms (using approximate values):

$$
\ln(0.175) \approx -1.743 \quad \text{and} \quad \ln(1.0075) \approx 0.00747
$$

Thus,

$$
n \approx -\frac{-1.743}{0.00747} \approx \frac{1.743}{0.00747} \approx 233.3
$$

Since the number of months must be a whole number and we round up, John will pay off the loan in:

$$
n = 234 \text{ months.}
$$
John will pay off the \$11,000 loan after 234 months.

Question 3
John takes out a loan of that charges interest compounded monthly. If John makes
monthly payments, determine how long it will take him to pay off the loan. Round your
answer up.
John will pay off the loan after
Time Value of Money Solver
Enter the given values.

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Question 3 John takes out a loan of that charges interest compounded monthly. If John makes monthly payments, determine how long it will take him to pay off the loan. Round your answer up. John will pay off the loan after Time Value of Money Solver Enter the given values.