Juan secured a 5-year car loan at 5.60% compounded annually that required her to make payments of $885.06 at the beginning of each month. Calculate the cost of the car if she made a downpayment of $1,750.
$0.00
Round to the nearest cent

To find the cost of the car, we need to determine the total amount paid over the life of the loan and then add the downpayment.

1. **Annual Interest Rate (r):** 5.60%
2. **Monthly Interest Rate (i):** $\frac{5.60\%}{12} = 0.0046667$
3. **Number of Payments (n):** 5 years $\times$ 12 months/year = 60 payments
4. **Monthly Payment (PMT):** $885.06

Since payments are made at the beginning of each month, this is an annuity due.

**Formula for the present value of an annuity due:**

\[ PV = PMT \times \left(1 + i\right) \times \left(1 - (1 + i)^{-n}\right) / i \]

Substituting the values:

\[ PV = 885.06 \times \left(1 + 0.0046667\right) \times \left(1 - (1 + 0.0046667)^{-60}\right) / 0.0046667 \]

Let's compute the present value step-by-step:

1. **Monthly Interest Rate Plus One:**
\[ 1 + 0.0046667 = 1.0046667 \]

2. **Negative Power Term:**
\[ (1.0046667)^{-60} \]

3. **Term Inside Parentheses:**
\[ 1 - (1.0046667)^{-60} \]

4. **Fraction Term:**
\[ \left(1 - (1.0046667)^{-60}\right) / 0.0046667 \]

5. **Multiplying by $ PMT $ and $ (1 + i) $:**
\[ 885.06 \times 1.0046667 \times \left(\frac{1 - (1.0046667)^{-60}}{0.0046667}\right) \]

Let's calculate this using a calculator:

1. **Negative Power Calculation:**
\[ (1.0046667)^{-60} \approx 0.7586 \]

2. **Term Inside Parentheses:**
\[ 1 - 0.7586 = 0.2414 \]

3. **Fraction Term:**
\[ \frac{0.2414}{0.0046667} \approx 51.729 \]

4. **Final Multiplication:**
\[ 885.06 \times 1.0046667 \times 51.729 \approx 45964.31 \]

**Present Value of Loan Payments (PV):** $45,964.31

**Total Cost of the Car:**
\[ \text{Total Cost} = \text{PV} + \text{Downpayment} = 45964.31 + 1750 = 47714.31 \]

**Rounded to the nearest cent:** $47,714.31

So, the cost of the car is $47,714.31.

Would you like more details or have any questions? Here are five additional questions you might ask:

1. How do you calculate the monthly interest rate from an annual rate?
2. What is the difference between an ordinary annuity and an annuity due?
3. How do different compounding periods affect the total interest paid?
4. Can you explain the steps to calculate the present value of an annuity?
5. What factors should be considered when choosing between different loan options?

**Tip:** When calculating the present value of annuities, always check if the payments are made at the beginning (annuity due) or the end (ordinary annuity) of the period, as this affects the formula used.

Learn how to calculate the total cost of a car financed with a 5-year loan at 5.60% annual interest rate compounded monthly, with monthly payments of $885.06 made at the beginning of each month. Includes step-by-step solution and formulas used in finance.

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