You want to buy equipment that is available from 2 companies. The price of the equipment is the same for both companies. Silver Sugar would let you make quarterly payments of $13,700 for 4 years at an interest rate of 2.64 percent per quarter. Your first payment to Silver Sugar would be in 3 months. River Sugar would let you make monthly payments of $3,272 for 5 years at an interest rate of X percent per month. Your first payment to River Sugar would be today. What is X?


We are given two different payment plans, and we need to find the monthly interest rate, $ X $, for River Sugar's plan. Here's the approach:

### Step 1: Calculate the present value of Silver Sugar's payments.

For Silver Sugar:
- Payments: Quarterly payments of $ 13,700 $
- Interest rate: $ 2.64\% $ per quarter (or $ 0.0264 $ as a decimal)
- Number of periods: 4 years, quarterly, which gives $ 4 \times 4 = 16 $ quarters
- Payments start in 3 months (first payment after one quarter)

The present value of the payments can be calculated using the formula for the present value of an ordinary annuity:

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

Where:
- $ P = 13,700 $ (the quarterly payment)
- $ r = 0.0264 $ (the quarterly interest rate)
- $ n = 16 $ (number of quarters)

Let’s calculate $ PV_{\text{Silver}} $.

### Step 2: Calculate the present value of River Sugar's payments.

For River Sugar:
- Payments: Monthly payments of $ 3,272 $
- Interest rate: $ X $ per month (which we need to find)
- Number of periods: 5 years, monthly, which gives $ 5 \times 12 = 60 $ months
- Payments start immediately (first payment today, this is an annuity due)

The present value of the payments can be calculated using the formula for the present value of an annuity due:

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

Where:
- $ P = 3,272 $ (the monthly payment)
- $ r = X $ (the monthly interest rate, which we need to find)
- $ n = 60 $ (number of months)

### Step 3: Equate the present values.

Since the price of the equipment is the same for both companies, we can equate the two present values:

\[
PV_{\text{Silver}} = PV_{\text{River}}
\]

This will allow us to solve for $ X $, the monthly interest rate for River Sugar.

Let's proceed with the calculations.The monthly interest rate, $ X $, for River Sugar is approximately $ 0.36\% $ per month.

Would you like more details on any part of the calculation or have additional questions?

Here are five related questions to deepen your understanding:
1. How does the timing of payments (immediate vs. delayed) affect the present value of payments?
2. What formula is used to calculate the present value of an annuity?
3. How would the total payment amount differ if the payments were not structured as an annuity?
4. How can the interest rate change if we increase the number of payments but keep the payment amount constant?
5. What impact does a higher interest rate have on the total cost of financing?

**Tip:** Always be mindful of whether payments are starting immediately or after a delay, as this impacts 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. Silver Sugar would let you make quarterly payments of $13,700 for 4 years at an interest rate of 2.64 percent per quarter. Your first payment to Silver Sugar would be in 3 months. River Sugar would let you make monthly payments of $3,272 for 5 years at an interest rate of X percent per month. Your first payment to River Sugar would be today. What is X?

Find the monthly interest rate for River Sugar's payment plan by calculating and equating the present value of annuities. Compare quarterly and monthly payment structures using key formulas in finance. Suitable for advanced high school math and undergraduate finance students.

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