You want to buy equipment that is available from 2 companies. The price of the equipment is the same for both companies. Gray Entertainment would let you make quarterly payments of $5,080 for 3 years at an interest rate of 2.60 percent per quarter. Your first payment to Gray Entertainment would be in 3 months. Oval Entertainment would let you make monthly payments of $X for 9 years at an interest rate of 0.85 percent per month. Your first payment to Oval Entertainment would be today. What is X?

To find the monthly payment $ X $ that you would make to Oval Entertainment, we need to compare the present value (PV) of both payment options, as the price of the equipment is the same.

### Step 1: Calculate the present value of payments to Gray Entertainment
Gray Entertainment offers quarterly payments of $ \$5,080 $ for 3 years at an interest rate of $ 2.60\% $ per quarter. The first payment is made in 3 months.

The formula for the present value of an ordinary annuity (where payments are made at the end of each period) is:

\[
PV = P \times \frac{1 - (1 + r)^{-n}}{r}
\]

Where:
- $ P = 5,080 $ (quarterly payment)
- $ r = 0.026 $ (quarterly interest rate)
- $ n = 3 \text{ years} \times 4 \text{ quarters per year} = 12 \text{ quarters} $

Substituting into the formula:

\[
PV = 5,080 \times \frac{1 - (1 + 0.026)^{-12}}{0.026}
\]

We can calculate this to find the present value of Gray Entertainment's offer.

### Step 2: Set up the equation for Oval Entertainment's monthly payments
Oval Entertainment offers monthly payments for 9 years at an interest rate of $ 0.85\% $ per month, and the first payment is made immediately (this is an annuity due). The present value of an annuity due is calculated by:

\[
PV = X \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \times (1 + r)
\]

Where:
- $ X $ is the unknown monthly payment
- $ r = 0.0085 $ (monthly interest rate)
- $ n = 9 \text{ years} \times 12 \text{ months per year} = 108 \text{ months} $

### Step 3: Equate the present values
Since the price of the equipment is the same for both companies, the present value of payments to Gray Entertainment should be equal to the present value of payments to Oval Entertainment. Therefore:

\[
PV_{\text{Gray}} = PV_{\text{Oval}}
\]

Substituting the expressions for both present values:

\[
5,080 \times \frac{1 - (1 + 0.026)^{-12}}{0.026} = X \times \left( \frac{1 - (1 + 0.0085)^{-108}}{0.0085} \right) \times (1 + 0.0085)
\]

Now we can calculate $ X $. Let's proceed with the calculations.The monthly payment $ X $ for Oval Entertainment would be approximately \$728.64.

Would you like further details or explanations? Here are five related questions to expand on this:

1. How does the concept of time value of money affect loan repayments?
2. What are the differences between an ordinary annuity and an annuity due?
3. How does increasing the interest rate impact the monthly payment amount?
4. What is the formula for calculating present value for different payment schedules?
5. How do loan terms (number of periods) influence total interest paid?

**Tip:** When comparing payment plans, always calculate the present value to understand the true cost of each option.

Solve a detailed payment comparison problem involving two companies with different interest rates and payment schedules. Learn about the time value of money, present value of annuities, and how to calculate equivalent monthly payments. Suitable for finance students or advanced high school math learners.

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