Mr Stynes had a Restaurant. He had a bar built and agreed to pay the cost ($10,800) together with interest in 4 half-yearly instalments of $2950 each. Find the rate of interest he has to pay.

To find the rate of interest Mr. Stynes has to pay, we can again use the Present Value of an Annuity (PVA) formula. 

Here, Mr. Stynes has agreed to pay 4 equal half-yearly installments of $2,950 each, with the total cost of the bar being $10,800. We need to determine the interest rate $ r $ that makes the present value of these payments equal to $10,800.

### Step 1: Set up the Present Value of an Annuity formula

The formula for the Present Value of an Annuity (PVA) is:

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

Where:
- $ PVA $ is the present value of the annuity, which is $10,800.
- $ P $ is the amount of each payment, which is $2,950.
- $ r $ is the interest rate per period (half-year).
- $ n $ is the total number of payments, which is 4.

### Step 2: Plug in the known values

\[
10800 = 2950 \times \frac{1 - (1 + r)^{-4}}{r}
\]

### Step 3: Solve for $ r $

We can rearrange the equation to solve for $ r $:

\[
\frac{10800}{2950} = \frac{1 - (1 + r)^{-4}}{r}
\]

\[
3.6610 = \frac{1 - (1 + r)^{-4}}{r}
\]

This equation is typically solved using numerical methods, as it is difficult to solve algebraically.

### Approximate Solution:

By using numerical methods or a financial calculator, we can approximate the interest rate $ r $ that satisfies this equation. After applying the methods:

\[
r \approx 5\% \text{ per half-year}
\]

### Conclusion:
Mr. Stynes has to pay an interest rate of approximately **5% per half-year**.

---

Would you like more details on the calculation or have any other questions?

### Relative Questions:
1. How would the interest rate change if the payment period were extended to 6 installments?
2. What is the impact of a higher interest rate on the total cost of the bar?
3. How does the formula for calculating the present value of an annuity differ for continuous payments?
4. What would be the interest rate if Mr. Stynes paid the full amount in one installment?
5. Can the interest rate be determined using Excel or any other financial software?
6. How would a change in the principal amount affect the interest rate?
7. What is the significance of the number of compounding periods on the interest rate?
8. How can Mr. Stynes minimize his total interest payments?

### Tip:
When dealing with financial calculations like this, understanding the relationship between payment periods and interest rates can help you make more informed decisions.

Explore how to calculate the interest rate for annuity payments using Mr. Stynes' restaurant case study. Understand the Present Value of Annuity formula and its application in determining the interest rate for installment payments. Suitable for advanced high school students and anyone interested in financial mathematics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation