You want to buy equipment that is available from 2 companies. The price of the equipment is the same for both companies. Silver Mining would let you make quarterly payments of $8,840 for 8 years at an interest rate of 3.42 percent per quarter. Your first payment to Silver Mining would be in 3 months. Lake Mining would let you make monthly payments of $4,105 for 5 years at an interest rate of X percent per month. Your first payment to Lake Mining would be today. What is X?
 

We need to determine the monthly interest rate $X$ for Lake Mining, given that the price of the equipment is the same from both companies. We'll use the concept of the present value of annuities to solve this, since both companies offer a structured payment plan with periodic payments.

### 1. Silver Mining's Plan:
- Quarterly payments of $8,840 for 8 years.
- Interest rate of 3.42% per quarter.
- Payments are made quarterly (every 3 months), with the first payment occurring after 3 months.

For Silver Mining, the present value $PV_{\text{Silver}}$ of these payments is given by the present value formula for an ordinary annuity:

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

Where:
- $P = 8,840$ is the quarterly payment.
- $r = 0.0342$ is the quarterly interest rate (3.42% per quarter).
- $n = 8 \times 4 = 32$ is the total number of payments (since 8 years with quarterly payments results in 32 payments).

### 2. Lake Mining's Plan:
- Monthly payments of $4,105 for 5 years.
- Interest rate is $X$ per month.
- Payments are made monthly (every month), with the first payment occurring immediately (this is an annuity due).

For Lake Mining, the present value $PV_{\text{Lake}}$ is given by the present value formula for an annuity due:

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

Where:
- $P = 4,105$ is the monthly payment.
- $r = X$ is the unknown monthly interest rate.
- $n = 5 \times 12 = 60$ is the total number of payments (since 5 years with monthly payments results in 60 payments).
- The extra factor $(1 + r)$ accounts for the annuity being due (first payment is immediate).

### 3. Setting the Present Values Equal:

Since the price of the equipment is the same from both companies, the present value of the two payment plans must be equal:

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

Substituting the formulas:

\[
8,840 \times \left( \frac{1 - (1 + 0.0342)^{-32}}{0.0342} \right) = 4,105 \times \left( \frac{1 - (1 + X)^{-60}}{X} \right) \times (1 + X)
\]

Now let's compute the left-hand side (the present value for Silver Mining), and then solve for $X$ on the right-hand side.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like to go over the steps manually, or do you have another question in mind? 

Here are 5 related questions you might find interesting:
1. How do you calculate the present value of an annuity due?
2. What is the difference between ordinary annuity and annuity due?
3. How can I calculate the monthly interest rate based on different payment schedules?
4. What are other ways companies structure payment plans with interest?
5. How does the time value of money affect purchasing decisions?

**Tip**: When comparing payment plans, always consider both the interest rate and the structure of the payments (monthly, quarterly, etc.) to evaluate the overall cost accurately.

You want to buy equipment that is available from 2 companies. The price of the equipment is the same for both companies. Silver Mining would let you make quarterly payments of $8,840 for 8 years at an interest rate of 3.42 percent per quarter. Your first payment to Silver Mining would be in 3 months. Lake Mining would let you make monthly payments of $4,105 for 5 years at an interest rate of X percent per month. Your first payment to Lake Mining would be today. What is X?

Understand how to calculate the monthly interest rate for a given equipment purchase, comparing quarterly payments with Silver Mining and monthly payments with Lake Mining. This problem involves concepts like the present value of annuities, interest rate calculations, and time value of money, suitable for college-level finance students.

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