You want to buy equipment that is available from 2 companies. The price of the equipment is the same for both companies. Red Construction would let you make quarterly payments of $13,570 for 6 years at an interest rate of 4.16 percent per quarter. Your first payment to Red Construction would be in 3 months. Valley Construction would let you make monthly payments of $X for 9 years at an interest rate of 1.04 percent per month. Your first payment to Valley Construction would be today. What is X?

To find the monthly payment $ X $ for Valley Construction, we need to calculate the present value of the payment plans offered by both Red Construction and Valley Construction and set them equal to each other since the total cost of the equipment is the same for both companies.

### Step 1: Present Value of Red Construction Payments
Red Construction offers quarterly payments of $ 13,570 $ for 6 years with an interest rate of 4.16% per quarter. The formula for the present value $ PV $ of an annuity is given by:

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

Where:
- $ P $ is the payment amount per period,
- $ r $ is the interest rate per period,
- $ n $ is the number of periods.

For Red Construction:
- $ P = 13,570 $,
- $ r = 0.0416 $ (4.16% per quarter),
- $ n = 6 \text{ years} \times 4 \text{ quarters/year} = 24 \text{ quarters} $.

Substituting these values into the formula:

\[
PV_{\text{Red}} = 13,570 \times \frac{1 - (1 + 0.0416)^{-24}}{0.0416}
\]

### Step 2: Present Value of Valley Construction Payments
Valley Construction offers monthly payments of $ X $ for 9 years with an interest rate of 1.04% per month. The present value $ PV $ for this plan is similarly calculated:

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

For Valley Construction:
- $ X = \text{monthly payment (to be determined)} $,
- $ r = 0.0104 $ (1.04% per month),
- $ n = 9 \text{ years} \times 12 \text{ months/year} = 108 \text{ months} $.

Substituting these values:

\[
PV_{\text{Valley}} = X \times \frac{1 - (1 + 0.0104)^{-108}}{0.0104}
\]

### Step 3: Set the Present Values Equal
Since the equipment costs the same from both companies, we set the present values equal:

\[
PV_{\text{Red}} = PV_{\text{Valley}}
\]

\[
13,570 \times \frac{1 - (1 + 0.0416)^{-24}}{0.0416} = X \times \frac{1 - (1 + 0.0104)^{-108}}{0.0104}
\]

We can now solve for $ X $. I'll calculate it now.The monthly payment $ X $ for Valley Construction would be approximately $ 3,146.15 $.

Would you like further details on the steps or concepts? Here are five related questions you might find interesting:

1. How would the monthly payment change if the interest rate for Valley Construction was lower?
2. What is the total amount paid over the 9 years to Valley Construction?
3. How does the timing of the first payment affect the present value in annuity calculations?
4. What would the effect be if Red Construction allowed monthly payments instead of quarterly?
5. Can you explain how compound interest impacts long-term payments in loan scenarios?

**Tip:** Understanding present value is crucial in comparing different payment plans as it adjusts future payments for the time value of money.

You want to buy equipment that is available from 2 companies. The price of the equipment is the same for both companies. Red Construction would let you make quarterly payments of $13,570 for 6 years at an interest rate of 4.16 percent per quarter. Your first payment to Red Construction would be in 3 months. Valley Construction would let you make monthly payments of $X for 9 years at an interest rate of 1.04 percent per month. Your first payment to Valley Construction would be today. What is X?

Learn how to determine monthly payments for Valley Construction over 9 years at an interest rate of 1.04% per month. This problem involves comparing two payment plans using present value formulas and annuity concepts. Suitable for college-level students studying finance or mathematics.

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