You want to buy equipment that is available from 2 companies. The price of the equipment is the same for both companies. Orange Furniture would let you make quarterly payments of $3,500 for 5 years at an interest rate of 4.27 percent per quarter. Your first payment to Orange Furniture would be in 3 months. Lake Furniture would let you make monthly payments of $X for 5 years at an interest rate of 1.41 percent per month. Your first payment to Lake Furniture would be today. What is X?



To solve this problem, we need to calculate the equivalent monthly payment $ X $ for Lake Furniture, given the quarterly payment structure of Orange Furniture. Here's the step-by-step breakdown:

### 1. Orange Furniture Payment Plan:
- Quarterly payment: \$3,500
- Interest rate: 4.27% per quarter
- Number of years: 5 years
- Total number of quarters: $ 5 \times 4 = 20 $
- The formula for the present value $ PV $ of an annuity (the total cost of the equipment) is:

\[
PV = P \times \frac{1 - (1 + r)^{-n}}{r}
\]
Where:
- $ P = 3,500 $ is the quarterly payment,
- $ r = \frac{4.27}{100} = 0.0427 $ is the interest rate per quarter,
- $ n = 20 $ is the number of quarterly payments.

### 2. Lake Furniture Payment Plan:
- Monthly payment: $ X $ (to be determined),
- Interest rate: 1.41% per month,
- Number of years: 5 years,
- Total number of months: $ 5 \times 12 = 60 $.

Since both companies charge the same total cost for the equipment, the present value of both payment plans must be equal. So, we need to set up a similar present value formula for Lake Furniture's monthly payments:

\[
PV = X \times \frac{1 - (1 + r_m)^{-n_m}}{r_m}
\]
Where:
- $ X $ is the monthly payment,
- $ r_m = \frac{1.41}{100} = 0.0141 $ is the monthly interest rate,
- $ n_m = 60 $ is the number of monthly payments.

Now, let's calculate $ PV $ for the Orange Furniture plan and solve for $ X $ for Lake Furniture.The monthly payment $ X $ for Lake Furniture would be approximately \$1,152.38.

Would you like further details or have any questions?

Here are five related questions you might find useful:
1. How is the present value formula for an annuity derived?
2. What is the impact of making payments at the beginning versus the end of each period?
3. How do changes in interest rates affect loan payment amounts?
4. Can we compare the total interest paid between Orange Furniture and Lake Furniture?
5. How would the solution change if the interest rates were compounded differently?

**Tip:** When comparing payment plans with different payment frequencies, always calculate the present value to make an accurate comparison.

You want to buy equipment that is available from 2 companies. The price of the equipment is the same for both companies. Orange Furniture would let you make quarterly payments of $3,500 for 5 years at an interest rate of 4.27 percent per quarter. Your first payment to Orange Furniture would be in 3 months. Lake Furniture would let you make monthly payments of $X for 5 years at an interest rate of 1.41 percent per month. Your first payment to Lake Furniture would be today. What is X?

Learn how to compare two payment plans for furniture purchase using annuity formulas. This step-by-step solution calculates monthly payments for Lake Furniture using a 1.41% monthly interest rate, given a quarterly payment plan from Orange Furniture. The problem explores the present value of annuities and interest rate calculations.

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