Use PMT $$ =\frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]} $$ to determine the regular payment amount, rounded to the nearest dollar. Your credit card has a balance of $$ \$ 5500 $$ and an annual interest rate of $$ 12 \% $$. You decide to pay off the balance over three years. If there are no further purchases charged to the card, a. How much must you pay each month? b. How much total interest will you pay? a. The monthly payments are approximately $$ \$ 183 $$. (Do not round until the final answer. Then round to the nearest dollar as needed.) b. The total interest paid over 3 years is approximately $$ \$ \square $$ (Round to the nearest dollar as needed.)

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UpStudy AI · Accepted Answer

Let

$$
P = 5500,\quad r = 0.12,\quad n = 12,\quad t = 3.
$$

The monthly interest rate is

$$
\frac{r}{n} = \frac{0.12}{12} = 0.01.
$$

The formula for the regular payment is

$$
\text{PMT} = \frac{P\left(\frac{r}{n}\right)}{1 - \left(1 + \frac{r}{n}\right)^{-nt}}.
$$

Substitute the known values:

$$
\text{PMT} = \frac{5500(0.01)}{1 - \left(1 + 0.01\right)^{-12 \times 3}} = \frac{55}{1 - (1.01)^{-36}}.
$$

Step 1. Compute $$(1.01)^{36}$$:

Taking natural logarithms, we have

$$
\ln(1.01) \approx 0.00995,
$$

so,

$$
\ln((1.01)^{36}) = 36 \cdot 0.00995 \approx 0.3582.
$$

Exponentiating gives

$$
(1.01)^{36} \approx e^{0.3582} \approx 1.4307.
$$

Thus,

$$
(1.01)^{-36} \approx \frac{1}{1.4307} \approx 0.699.
$$

Step 2. Compute the denominator:

$$
1 - (1.01)^{-36} \approx 1 - 0.699 = 0.301.
$$

Step 3. Calculate the payment:

$$
\text{PMT} \approx \frac{55}{0.301} \approx 182.724.
$$

Rounded to the nearest dollar, the monthly payment is

$$
\boxed{\$183}.
$$

For part (b), the total amount paid over 3 years (36 months) is

$$
183 \times 36 = 6588.
$$

The total interest paid is the total payment minus the principal:

$$
\text{Interest} = 6588 - 5500 = 1088.
$$

Thus, the total interest paid over 3 years, rounded to the nearest dollar, is

$$
\boxed{\$1088}.
$$
a. You must pay approximately \$183 each month. b. The total interest paid over 3 years is approximately \$1,088.

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