Annuity formula.
Brian Bell & Co. is selling micro-oven for K4000 by cash. If this item is purchased on an
installments basis where the method of repayment is to deposit K500 followed by equal semiannually repayments of K900 for three years.
(a) Calculate the flat rate of interest charged if this item is bought on an installments basis.
(b) Calculate the total amount paid at the end of 3 years if this item is bought on installments 
basis.


Let's break down the two parts of the problem and solve them step by step.

### (a) **Calculate the flat rate of interest charged if the item is bought on an installment basis.**

We are given:
- **Cash price**: K4,000 (price of the micro-oven)
- **Down payment**: K500
- **Semi-annual repayments**: K900 (every 6 months for 3 years)
- **Number of payments**: \( 3 \, \text{years} \times 2 \, \text{payments per year} = 6 \, \text{payments} \)

#### Total Amount Paid (Installments):
Since the installments are K900 each and there are 6 payments over 3 years, the total amount paid in installments is:
\[
\text{Total Installments Paid} = K900 \times 6 = K5,400
\]

The buyer also made an initial down payment of K500, so the total amount paid is:
\[
\text{Total Amount Paid} = K500 + K5,400 = K5,900
\]

#### Flat Interest Rate:
The flat interest is calculated on the difference between the cash price and the total amount paid.

\[
\text{Interest} = \text{Total Amount Paid} - \text{Cash Price} = K5,900 - K4,000 = K1,900
\]

The flat rate of interest is expressed as a percentage of the cash price:
\[
\text{Flat Rate of Interest} = \left(\frac{\text{Interest}}{\text{Cash Price}}\right) \times 100 = \left(\frac{1,900}{4,000}\right) \times 100 = 47.5\%
\]

So, the flat rate of interest charged is **47.5%**.

---

### (b) **Calculate the total amount paid at the end of 3 years if bought on installments.**

This part has been solved in part (a). The total amount paid consists of:
- **Down payment**: K500
- **Total installment payments**: K5,400

Therefore, the total amount paid over the 3-year period is:
\[
\text{Total Amount Paid} = K500 + K5,400 = K5,900
\]

---

### Summary of Results:
- **(a) Flat rate of interest charged**: **47.5%**
- **(b) Total amount paid at the end of 3 years**: **K5,900**

---

Would you like more details or clarifications? Here are some follow-up questions to deepen understanding:

1. How would the calculations change if the semi-annual payment was K1,000 instead of K900?
2. What is the effective annual interest rate if compounded semi-annually?
3. How would the total interest change if the repayment period was extended to 4 years?
4. What would be the flat rate if only K1,000 was paid upfront instead of K500?
5. How is flat interest different from compound interest in installment purchases?

**Tip:** Flat interest is easy to calculate but doesn't account for the decreasing loan balance as payments are made, which results in a higher effective interest rate than compound interest.

Brian Bell & Co. is selling micro-oven for K4000 by cash. If this item is purchased on an installments basis where the method of repayment is to deposit K500 followed by equal semi-annually repayments of K900 for three years. (a) Calculate the flat rate of interest charged if this item is bought on an installments basis. (b) Calculate the total amount paid at the end of 3 years if this item is bought on installments basis.

Learn how to calculate the flat rate of interest and the total amount paid over three years for a K4000 micro-oven bought on an installment basis. The solution includes key formulas for simple interest and installment payments, covering important financial mathematics concepts. Suitable for students in Grades 10-12.

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