QUESTION 3 SA Credit Bank Mia Parker P. O. Box 472 Kensington Maitland 7405 Dear Ms Parker We know it is important to you to manage your finances responsibly. Here's an offer you would want to use. R9 000,00 Cash already approved YOUR LOAN OFFER AMOUNT ALREADY APPROVED R 9000,00 - Payable over 48 months - Monthly installments R 318,92 EXPIRY DATE: 16 January Calculate the TOTAL amount that Mia has to pay back if she takes the loan. Why, do you think, do banks and other financial institutions offer cash loans to people that did not apply for it? Which investment is the most profitable? Show al calculations. (i) R560 invested at $$ 8 \% $$ R560 invested at p.a. simple interest for 3 years OR (ii) $$ \begin{array}{l} \text { R } 560 \text { invested at } 7 \% \ \text { p.a. compound } \end{array} $$ (2) .2 $$ \begin{array}{l} \text { R sou compound } \ \text { p.a. } \ \text { interest for } 3 \text { years } \end{array} $$

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Let's break down the problem step by step.

### Part 1: Calculate the Total Amount to Pay Back

Mia Parker has been offered a loan of R9,000, which she will pay back over 48 months with monthly installments of R318.92.

1. **Calculate the Total Amount Paid Back:**
   $$
   \text{Total Amount} = \text{Monthly Installment} \times \text{Number of Months}
   $$
   $$
   \text{Total Amount} = R318.92 \times 48
   $$

Now, let's calculate this value.

### Part 2: Why Do Banks Offer Cash Loans?

Banks and financial institutions often offer cash loans to individuals who did not apply for them as a marketing strategy. They may do this to:
- Attract new customers by providing them with easy access to credit.
- Increase their loan portfolio and interest income.
- Encourage spending, which can lead to more deposits and business for the bank.
- Build customer loyalty, hoping that the borrower will return for future financial products.

### Part 3: Compare Investments

We need to compare two investment options:

1. **Option (i): Simple Interest**
   - Principal (P) = R560
   - Rate (r) = 8% per annum
   - Time (t) = 3 years

The formula for simple interest is:
   $$
   \text{Simple Interest} = P \times r \times t
   $$
   The total amount after 3 years will be:
   $$
   \text{Total Amount} = P + \text{Simple Interest}
   $$

2. **Option (ii): Compound Interest**
   - Principal (P) = R560
   - Rate (r) = 7% per annum
   - Time (t) = 3 years

The formula for compound interest is:
   $$
   A = P \left(1 + \frac{r}{100}\right)^t
   $$
   where $$A$$ is the total amount after time $$t$$.

Now, let's perform the calculations for both parts.

### Calculations

1. **Total Amount to Pay Back for the Loan:**
   $$
   \text{Total Amount} = R318.92 \times 48
   $$

2. **Simple Interest Calculation:**
   $$
   \text{Simple Interest} = 560 \times 0.08 \times 3
   $$
   $$
   \text{Total Amount} = 560 + \text{Simple Interest}
   $$

3. **Compound Interest Calculation:**
   $$
   A = 560 \left(1 + \frac{7}{100}\right)^3
   $$

Let's calculate these values.
Calculate the value by following steps:
- step0: Calculate:
  $$318.92\times 48$$
- step1: Multiply the numbers:
  $$15308.16$$
Calculate or simplify the expression $$ 560 * 0.08 * 3 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$560\times 0.08\times 3$$
- step1: Multiply the terms:
  $$44.8\times 3$$
- step2: Multiply the numbers:
  $$134.4$$
Calculate or simplify the expression $$ 560 * (1 + 0.07)^3 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$560\left(1+0.07\right)^{3}$$
- step1: Add the numbers:
  $$560\times 1.07^{3}$$
- step2: Convert the expressions:
  $$560\left(\frac{107}{100}\right)^{3}$$
- step3: Simplify:
  $$560\times \frac{107^{3}}{100^{3}}$$
- step4: Rewrite the expression:
  $$20\times 28\times \frac{107^{3}}{100^{3}}$$
- step5: Rewrite the expression:
  $$20\times 28\times \frac{107^{3}}{20^{3}\times 5^{3}}$$
- step6: Reduce the numbers:
  $$28\times \frac{107^{3}}{20^{2}\times 5^{3}}$$
- step7: Rewrite the expression:
  $$4\times 7\times \frac{107^{3}}{20^{2}\times 5^{3}}$$
- step8: Rewrite the expression:
  $$4\times 7\times \frac{107^{3}}{4^{2}\times 5^{2}\times 5^{3}}$$
- step9: Reduce the numbers:
  $$7\times \frac{107^{3}}{4\times 5^{2}\times 5^{3}}$$
- step10: Multiply:
  $$\frac{7\times 107^{3}}{12500}$$
### Results of the Calculations

1. **Total Amount to Pay Back for the Loan:**
   $$
   \text{Total Amount} = R318.92 \times 48 = R15,308.16
   $$

2. **Simple Interest Calculation:**
   $$
   \text{Simple Interest} = 560 \times 0.08 \times 3 = R134.40
   $$
   $$
   \text{Total Amount} = 560 + 134.40 = R694.40
   $$

3. **Compound Interest Calculation:**
   $$
   A = 560 \left(1 + 0.07\right)^3 \approx R686.02
   $$

### Summary of Findings

- **Total Amount to Pay Back for the Loan:** R15,308.16
- **Total Amount from Simple Interest Investment:** R694.40
- **Total Amount from Compound Interest Investment:** R686.02

### Conclusion

- The total amount Mia has to pay back if she takes the loan is **R15,308.16**.
- The most profitable investment is the simple interest option, yielding **R694.40**, compared to the compound interest option, which yields **R686.02**.
- **Total Amount to Pay Back for the Loan:** R15,308.16 - **Most Profitable Investment:** R560 invested at 8% simple interest for 3 years yields R694.40, which is more than the R686.02 from compound interest.

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