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.

This problem involves finding the rate of interest for an instalment payment plan. Let's use the formula for the amount paid in an instalment loan, which is typically modeled by the annuity formula. The general form for the present value of an annuity is:

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

Where:
- $ P $ is the present value (the loan amount, $10,800).
- $ R $ is the amount of each instalment ($2950).
- $ i $ is the interest rate per period (which we need to find).
- $ n $ is the number of periods (in this case, 4 half-yearly periods).

We need to solve for $ i $, the interest rate per period. Rearranging the formula and using numerical methods will help to determine $ i $.

Let me compute that for you.The solution for the interest rate is very complex, as it involves nested radicals. To simplify this, we can use numerical methods to approximate the value of $ i $, the interest rate per period. Let me calculate the numerical approximation.The interest rate per half-year is approximately **3.64%**. Therefore, the annual interest rate he has to pay is about **7.28%**.

Would you like any further details or have any questions?

Here are 5 related questions to expand your understanding:
1. How would the total payment change if the interest rate increased by 1% annually?
2. What happens to the instalment amounts if he extends the payment period to 6 instalments?
3. How can you calculate the total interest paid over the entire period?
4. What is the formula for calculating interest in a lump-sum payment situation instead of instalments?
5. How does compound interest affect long-term loans differently than simple interest?

**Tip:** When calculating interest, always ensure the interest rate matches the period used (half-yearly, quarterly, etc.).

Learn how to calculate the interest rate for a loan of $10,800 paid in 4 half-yearly instalments of $2950 each. This solution covers the annuity formula, interest calculation, and numerical methods to find the interest rate. Suitable for students in Grades 11-12 or college.

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