You want to buy equipment that is available from 2 companies. The price of the equipment is the same for both companies. Gray Fashion would let you make quarterly payments of $11,470 for 3 years at an interest rate of 1.29 percent per quarter. Your first payment to Gray Fashion would be in 3 months. Square Fashion would let you make monthly payments of $X for 4 years at an interest rate of 0.58 percent per month. Your first payment to Square Fashion would be today. What is X?



We can solve this problem by calculating the present value (PV) of the payment plans for both companies, since the price of the equipment is the same for both. The approach is to compute the present value of the payment stream for both Gray Fashion and Square Fashion, equate them, and solve for $ X $, the monthly payment for Square Fashion.

### Step 1: Gray Fashion's Payment Plan
Gray Fashion offers quarterly payments of $11,470 for 3 years, at an interest rate of 1.29% per quarter. The first payment is made in 3 months. 

We use the formula for the present value of an ordinary annuity (since the first payment occurs in 3 months, which makes it an ordinary annuity):

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

Where:
- $ P = 11,470 $ (quarterly payment),
- $ r = 0.0129 $ (quarterly interest rate),
- $ n = 3 \times 4 = 12 $ (number of quarterly payments).

### Step 2: Square Fashion's Payment Plan
Square Fashion offers monthly payments of $ X $ for 4 years, at an interest rate of 0.58% per month. The first payment is made today, so this is an annuity due.

The formula for the present value of an annuity due is:

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

Where:
- $ r = 0.0058 $ (monthly interest rate),
- $ n = 4 \times 12 = 48 $ (number of monthly payments),
- $ X $ is the monthly payment we want to solve for.

### Step 3: Equating the Present Values
Since the present value of both payment plans must be equal (because the equipment price is the same), we can equate the two present value equations:

\[
P \times \frac{1 - (1 + 0.0129)^{-12}}{0.0129} = X \times \left( \frac{1 - (1 + 0.0058)^{-48}}{0.0058} \right) \times (1 + 0.0058)
\]

We can now compute each side to solve for $ X $. Let's calculate this.The monthly payment $ X $ for Square Fashion would be approximately \$3,015.62.

Would you like further details on the steps or calculations? Here are some related questions you might find helpful:

1. How does the interest rate affect the present value of the payments?
2. What is the difference between an ordinary annuity and an annuity due?
3. How would the payment change if the interest rate for Square Fashion were higher?
4. How do you calculate present value for irregular payment intervals?
5. What is the formula for calculating the future value of these payments?

**Tip:** An annuity due has payments starting immediately, whereas an ordinary annuity begins after one period, which affects the present value calculation.

You want to buy equipment that is available from 2 companies. The price of the equipment is the same for both companies. Gray Fashion would let you make quarterly payments of $11,470 for 3 years at an interest rate of 1.29 percent per quarter. Your first payment to Gray Fashion would be in 3 months. Square Fashion would let you make monthly payments of $X for 4 years at an interest rate of 0.58 percent per month. Your first payment to Square Fashion would be today. What is X?

Learn how to calculate the monthly payment for an equipment purchase using present value and annuity formulas. This problem involves quarterly payments at 1.29% interest from Gray Fashion versus monthly payments from Square Fashion at 0.58% interest. The first payment to Gray is in 3 months, while Square's starts immediately. Suitable for advanced high school or college finance students.

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