John takes out a 6 year loan for $$ \$ 9100 $$ at $$ 11 \% $$ interest compounded monthly. Calculate his monthly payment. John's monthly payment will be $$ \$ \square $$

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UpStudy AI · Accepted Answer

We start with the formula for the monthly payment on a compound interest loan:

$$
M = \frac{rP}{1 - (1 + r)^{-n}}
$$

where:
- $$M$$ is the monthly payment,
- $$P$$ is the principal amount, here $$ \$9100 $$,
- $$r$$ is the monthly interest rate,
- $$n$$ is the total number of monthly payments.

**Step 1. Compute the monthly interest rate $$r$$:**

The annual interest rate is $$ 11\% $$, so

$$
r = \frac{0.11}{12} \approx 0.00916667.
$$

**Step 2. Compute the total number of monthly payments $$n$$:**

The loan term is 6 years, so

$$
n = 6 \times 12 = 72.
$$

**Step 3. Substitute the values into the formula:**

$$
M = \frac{0.00916667 \times 9100}{1 - (1 + 0.00916667)^{-72}}.
$$

**Step 4. Calculate the numerator:**

$$
0.00916667 \times 9100 \approx 83.41667.
$$

**Step 5. Calculate the denominator:**

First, we find $$ (1 + r)^{72} $$:

$$
(1 + 0.00916667)^{72} \approx 1.93.
$$

Taking the reciprocal:

$$
(1 + 0.00916667)^{-72} \approx \frac{1}{1.93} \approx 0.519.
$$

Thus, the denominator is:

$$
1 - 0.519 = 0.481.
$$

**Step 6. Compute the monthly payment $$M$$:**

$$
M \approx \frac{83.41667}{0.481} \approx 173.48.
$$

John's monthly payment will be $$ \$173.48 $$.
John's monthly payment will be \$173.48.

John takes out a 6 year loan for at interest compounded monthly. Calculate his
monthly payment.
John’s monthly payment will be

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John takes out a 6 year loan for at interest compounded monthly. Calculate his monthly payment. John’s monthly payment will be