You want to buy equipment that is available from 2 companies. The price of the equipment is the same for both companies. Red Technology would let you make quarterly payments of $7,910 for 4 years at an interest rate of 2.70 percent per quarter. Your first payment to Red Technology would be in 3 months. Circle Technology would let you make monthly payments of $X for 5 years at an interest rate of 1.18 percent per month. Your first payment to Circle Technology would be today. What is X?

We need to calculate the monthly payment $ X $ for Circle Technology. We are given information for two companies offering equipment financing with the same price but different payment schedules. Here's how we can solve for $ X $:

### 1. Red Technology:
- Quarterly payments: $7,910
- Payment term: 4 years (i.e., 16 quarterly payments)
- Interest rate: 2.70% per quarter

Since Red Technology offers a deferred annuity with equal quarterly payments, we can use the **present value of an ordinary annuity formula** to determine the equipment price:

\[
PV = P \times \left( \frac{1 - (1 + r)^{-n}}{r} \right)
\]
Where:
- $ PV $ is the present value (equipment price),
- $ P = 7,910 $ is the quarterly payment,
- $ r = 0.027 $ (2.70% per quarter),
- $ n = 16 $ (16 quarterly payments).

Let's calculate the present value (price) based on Red Technology’s plan first.

### 2. Circle Technology:
- Monthly payments: $ X $ (unknown)
- Payment term: 5 years (i.e., 60 monthly payments)
- Interest rate: 1.18% per month
- Payments begin immediately (annuity due)

For Circle Technology, we use the **present value of an annuity due** formula:
\[
PV = X \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \times (1 + r)
\]
Where:
- $ PV $ is the same present value as calculated for Red Technology,
- $ X $ is the monthly payment,
- $ r = 0.0118 $ (1.18% per month),
- $ n = 60 $ (60 monthly payments).

We will first calculate the present value $ PV $ for Red Technology and then solve for $ X $ using the same $ PV $ in the Circle Technology formula.The equipment price (present value) is approximately \$101,675.89 based on Red Technology's financing plan. The monthly payment $ X $ for Circle Technology would be approximately \$2,346.57.

Would you like more details or have any questions? 

Here are five related questions to expand your understanding:
1. How does the interest rate affect the total amount paid over time?
2. What are the key differences between an annuity due and an ordinary annuity?
3. How would the payments change if the interest rates were compounded differently?
4. What happens to the payment amounts if you extend or shorten the payment period?
5. How do you calculate the total amount paid in both financing options?

**Tip:** Always consider the difference in timing for payments when comparing annuity options, as starting payments earlier (like in Circle Technology) impacts the overall interest paid.

This problem demonstrates how to calculate monthly payments for Circle Technology based on a given annuity due formula. Learn key financial math concepts such as present value of annuities, interest rate effects, and payment schedules for equipment financing. Suitable for undergraduate finance students or advanced high school math learners.

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