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 formula for the payment on an amortized loan:

$$
\text{Payment} = \frac{P i}{1 - \left(1+i\right)^{-n}},
$$

where  
$$ P $$ is the principal loan amount,  
$$ i $$ is the monthly interest rate, and  
$$ n $$ is the number of monthly payments.

**Step 1. Identify the given values.**

$$
P = 11000, \quad \text{Annual interest rate} = 9\% \quad \Longrightarrow \quad i = \frac{0.09}{12} = 0.0075,
$$
$$
\text{Monthly Payment} = 100.
$$

**Step 2. Write the payment formula with the values.**

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

**Step 3. Compute the numerator.**

$$
11000 \times 0.0075 = 82.5.
$$

So the equation becomes:

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

**Step 4. Solve for $$1 - (1.0075)^{-n}$$.**

Multiply both sides by the denominator:

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

Divide both sides by 100:

$$
1 - \left(1.0075\right)^{-n} = \frac{82.5}{100} = 0.825.
$$

Then, isolate the exponential term:

$$
\left(1.0075\right)^{-n} = 1 - 0.825 = 0.175.
$$

**Step 5. Solve for $$n$$ using logarithms.**

Take the natural logarithm of both sides:

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

Using the logarithm rule $$\ln(a^b) = b \ln(a)$$, we have:

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

Now solve for $$n$$:

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

**Step 6. Compute the value of $$n$$.**

Calculate the logarithms (using a calculator):

$$
\ln(0.175) \approx -1.743668,
$$
$$
\ln(1.0075) \approx 0.007471.
$$

Thus,

$$
n \approx -\frac{-1.743668}{0.007471} \approx \frac{1.743668}{0.007471} \approx 233.36.
$$

Since $$n$$ must be a whole number of months and we round up to ensure the loan is fully repaid,

$$
n = 234 \text{ months}.
$$

John will pay off the $$\$11000$$ loan after $$234$$ 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.