### 1. Topic Overview

**Main Concept/Theme:** Financial Mathematics involves understanding and applying mathematical principles to solve problems related to finance such as interest calculations, inflation, annuities, loans, and investments.

**Key Learning Objectives:**

– Understand different types of interest (simple and compound).

– Calculate present and future values of annuities.

– Analyze the effect of inflation on financial decisions.

– Solve problems involving loans and monthly repayments.

### 2. Key Terms and Definitions

**Simple Interest:**Interest calculated on the principal balance only.**Compound Interest:**Interest calculated on the initial principal and also on the accumulated interest of previous periods.**Principal:**The initial amount of money invested or borrowed.**Annuity:**A series of equal payments made at regular intervals over a period of time.**Future Value:**The value of an investment at a specified date in the future.**Present Value:**The current value of a future amount of money or stream of cash flows given a specified rate of return.**Inflation:**The rate at which the general level of prices for goods and services is rising.**Interest Rate:**The proportion of a loan that is charged as interest to the borrower.**Amortization:**The process of spreading out a loan into a series of fixed payments.

### 3. Main Content Sections

#### Simple Interest

**Formula:**

[I = P \times r \times t]

Where:

– (I) is the interest

– (P) is the principal amount

– (r) is the annual interest rate

– (t) is the time in years

**Example:**

If (P = R1000), (r = 5\%), and (t = 2) years:

[I = 1000 \times 0.05 \times 2 = R100]

#### Compound Interest

**Formula:**

[A = P \left(1 + \frac{r}{n}\right)^{nt}]

Where:

– (A) is the amount of money accumulated after (n) periods

– (P) is the principal

– (r) is the annual interest rate

– (n) is the number of times that interest is compounded per unit time

– (t) is the time the money is invested for

**Example:**

If (P = R1000), (r = 5\%), (t = 2) years, and compounded annually ((n = 1)):

[A = 1000 \left(1 + \frac{0.05}{1}\right)^{1 \times 2} = 1000 \left(1.05\right)^2 = R1102.50]

#### Future and Present Values of Annuities

**Future Value of an Annuity Formula:**

[ FV = P \left(\frac{(1 + r)^n – 1}{r}\right) ]

**Present Value of an Annuity Formula:**

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

Where:

– (P) is the payment amount

– (r) is the interest rate per period

– (n) is the number of payments

#### Loans and Amortization

**Loan Repayment Formula:**

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

Where:

– (R) is the monthly repayment amount

– (P) is the loan amount

– (r) is the monthly interest rate

– (n) is the total number of repayments

### 4. Example Problems or Case Studies

**Case Study 1:**

Sarah invests R5000 at an interest rate of 6% compounded annually for 3 years. Calculate the future value of her investment.

[A = 5000 \left(1 + \frac{0.06}{1}\right)^{1 \times 3} = 5000 \left(1.06\right)^3 = R5955.08]

**Case Study 2:**

John wants to buy a car worth R200,000. He plans to take a loan to be repaid in 5 years with an annual interest rate of 8% compounded monthly. Calculate his monthly repayments.

[ R = \frac{200000 \cdot 0.00667 \cdot (1 + 0.00667)^{60}}{(1 + 0.00667)^{60} – 1} \approx R4055.40 ]

### 5. Summary or Review Section

- Simple interest is calculated on the principal only, while compound interest is calculated on the principal and accrued interest.
- Future and present values help in determining the worth of investments and payments over time.
- Loans have fixed repayments, including principal and interest, calculated using the amortization formula.
- Understanding inflation is crucial for making long-term financial decisions.

### 6. Self-Assessment Questions

- Define simple interest and compound interest.
- Calculate the interest on a R15,000 investment at an annual simple interest rate of 7% for 4 years.
- If an amount of R8,000 is compounded semi-annually at a rate of 10% for 3 years, what will be the accumulated amount?
- Find the future value of an annuity with monthly payments of R1,000 for 5 years at an annual interest rate of 5%.
- A loan of R150,000 is to be repaid in 10 years with monthly installments at an annual interest rate of 6% compounded monthly. Calculate the monthly repayment amount.

### 7. Connections to Other Topics/Subjects

**Economics:**Understanding the time value of money and inflation’s impact on purchasing power.**Accounting:**Applying interest calculations in financial reporting and auditing.**Business Studies:**Evaluating investment opportunities and financial planning.**Life Skills:**Personal financial management and decision-making for future investments.

Remember to review these notes regularly and practice solving problems to strengthen your understanding of financial mathematics. If you need help, don’t hesitate to ask your teacher or peers for clarification.