Shamut Construction Company takes out a loan for $25,000 to pay for some new equipment. They decide to make quarterly payments and pay off the loan in 3 years. If the bank charges the company an annual interest rate of 3%, how much are their quarterly payments?

To calculate the quarterly payments for the loan, we can use the formula for the payment on an amortized loan:

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

Where:
- $ P $ is the quarterly payment.
- $ PV $ is the loan amount (the present value), which is \$25,000.
- $ r $ is the quarterly interest rate (annual interest rate divided by 4).
- $ n $ is the total number of payments (number of quarters over 3 years).

### Step 1: Calculate the Quarterly Interest Rate
The annual interest rate is 3%, so the quarterly interest rate is:
\[
r = \frac{0.03}{4} = 0.0075
\]

### Step 2: Calculate the Total Number of Payments
Since the loan is to be paid off in 3 years with quarterly payments, the total number of payments is:
\[
n = 3 \times 4 = 12
\]

### Step 3: Plug Values into the Formula
Now we can substitute the values into the payment formula:

\[
P = \frac{0.0075 \times 25,000}{1 - (1 + 0.0075)^{-12}}
\]

### Step 4: Simplify the Expression
First, calculate $ (1 + 0.0075)^{-12} $:
\[
(1 + 0.0075)^{-12} = (1.0075)^{-12} \approx 0.91057
\]

Now, calculate the denominator:
\[
1 - 0.91057 = 0.08943
\]

Finally, calculate the payment:
\[
P = \frac{0.0075 \times 25,000}{0.08943} = \frac{187.5}{0.08943} \approx 2,096.83
\]

### Result:
The quarterly payment is approximately **\$2,096.83**.

Would you like further clarification on any step? Here are some related questions you might find helpful:

1. What would the payments be if the interest rate were 5% instead?
2. How much total interest will Shamut Construction pay over the 3 years?
3. What is the remaining balance after 1 year of payments?
4. How would the payment change if they took 5 years to pay off the loan instead of 3?
5. What if they made monthly payments instead of quarterly?

**Tip:** Understanding the effect of interest rate and loan period can help in making better financial decisions when borrowing.

Find out how to calculate quarterly payments for a $25,000 loan over 3 years at an annual interest rate of 3%. This detailed solution explains key concepts such as amortization, interest rates, and loan payment formulas, suitable for students in Grades 10-12.

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