You want to buy equipment that is available from 2 companies. The price of the equipment is the same for both companies. Pink Research would let you make quarterly payments of $19,360 for 4 years at an interest rate of 3.74 percent per quarter. Your first payment to Pink Research would be in 3 months. Square Research would let you make monthly payments of $X for 6 years at an interest rate of 1.30 percent per month. Your first payment to Square Research would be today. What is X?

 



This problem involves comparing two different payment options offered by Pink Research and Square Research, using the concept of the present value of annuities. To find the monthly payment $X$ for Square Research, we will need to follow these steps:

### Step 1: Calculate the present value of the Pink Research option.
The price of the equipment is not directly given but can be determined using the quarterly payments to Pink Research. We'll use the formula for the present value of an annuity:

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

Where:
- $P = 19,360$ is the quarterly payment.
- $r = 3.74\%$ (or 0.0374) is the quarterly interest rate.
- $n = 4 \times 4 = 16$ is the total number of payments (since there are 4 years and 4 payments per year).

We will calculate the present value of these quarterly payments.

### Step 2: Set up the equation for Square Research.
For Square Research, the same present value $PV$ (from Step 1) applies. Now, we need to find $X$, the monthly payment. The relevant formula is:

\[
PV = X \times \left[ 1 - \left(1 + r_m\right)^{-n_m} \right] \div r_m
\]

Where:
- $r_m = 1.30\%$ (or 0.013) is the monthly interest rate.
- $n_m = 6 \times 12 = 72$ is the total number of monthly payments.
- $X$ is the monthly payment we need to find.

Now, we can solve for $X$ by equating the present values from both companies and rearranging the equation.

Let's calculate the values step by step.The present value of the equipment based on Pink Research's payment plan is approximately $ 229,977.54 $. 

For Square Research, the monthly payment $ X $ would be approximately $ 4,938.13 $.

Would you like further details or clarifications on any part of this solution? Here are some additional questions to explore:

1. How would the solution change if the interest rates were slightly different for both companies?
2. What is the total amount paid to each company after all payments are made?
3. How does changing the payment period affect the overall cost of the equipment?
4. What if you could make a lump-sum payment? How would you calculate that?
5. How would inflation impact the decision between these two options?

**Tip:** When comparing loans, always calculate the total cost by summing up all payments, not just the interest rate.

You want to buy equipment that is available from 2 companies. The price of the equipment is the same for both companies. Pink Research would let you make quarterly payments of $19,360 for 4 years at an interest rate of 3.74 percent per quarter. Your first payment to Pink Research would be in 3 months. Square Research would let you make monthly payments of $X for 6 years at an interest rate of 1.30 percent per month. Your first payment to Square Research would be today. What is X?

Learn how to compare two payment plans for an equipment purchase using the present value of annuities formula. Pink Research offers quarterly payments over 4 years at 3.74% interest, while Square Research offers monthly payments over 6 years at 1.30%. Find out how to calculate the monthly payment (X) and understand the time value of money.

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