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 tasked with finding the monthly interest rate $ X \% $ for River Sugar, where the monthly payments equal the quarterly payment plan of Silver Sugar.

### Step 1: Understand the structure of the problem

We are given:
- Silver Sugar offers quarterly payments of \$13,700 for 4 years, with an interest rate of 2.64% per quarter.
- River Sugar offers monthly payments of \$3,272 for 5 years, with an unknown monthly interest rate $ X \% $, and the first payment is made today.

Our goal is to find $ X $, the monthly interest rate for River Sugar, such that both payment plans have the same present value.

### Step 2: Calculate the present value (PV) for Silver Sugar

The formula for the present value of an annuity is:

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

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

For Silver Sugar:
- $ P = 13,700 $ (quarterly payment),
- $ r = \frac{2.64}{100} = 0.0264 $ (quarterly interest rate),
- $ n = 4 \times 4 = 16 $ (since there are 4 years with 4 quarters each).

Now, we can compute the present value for Silver Sugar.

### Step 3: Calculate the present value for River Sugar

For River Sugar:
- $ P = 3,272 $ (monthly payment),
- $ n = 5 \times 12 = 60 $ (since there are 5 years with 12 months each),
- $ r $ is unknown, and we need to solve for $ r $.

We know that the present value must be equal for both payment plans, so we set the present values equal to each other:

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

Let's now calculate these values.It seems like I can’t do more advanced data analysis right now. Please try again later. If you'd like, I can guide you through the steps to calculate the present value manually or provide some alternative suggestions.

Would you like more details on the formula and approach? Let me know if you have any questions!

Here are 5 related questions:
1. How does the concept of present value apply to other financial decisions?
2. What is the difference between monthly and quarterly payment structures?
3. How do interest rates affect the total cost of payment plans?
4. What other factors should you consider when choosing between payment plans?
5. What are some advantages of paying upfront instead of installments?

**Tip**: For financial calculations, always double-check the period (monthly, quarterly) when comparing loans or investment options to avoid confusion over the true interest rate.

This problem compares two payment plans for buying equipment, using the present value formula for annuities. Silver Sugar offers quarterly payments, while River Sugar offers monthly payments. Learn how to calculate the monthly interest rate that makes these two plans equivalent using the present value formula.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation