Future Value Annuities (Grade 12 NSC Matric Mathematics): Revision Notes

Worked Example: Basic Future Value Calculation

Question: Kobus deposits R500 at the end of each year for 4 years into an investment account with 10% interest per annum compounded yearly. Find the value of his investment after 4 years.

Solution:

Step 1: Write down the given information and formula

• $F = x\frac{(1 + i)^n - 1}{i}$
• $x = 500$
• $i = 0.1$
• $n = 4$

Step 2: Draw a timeline to visualise the problem

The timeline shows deposits of R500 at T₁, T₂, T₃, and T₄, with 10% compound interest.

Step 3: Apply the formula

• $F = 500\frac{(1 + 0.1)^4 - 1}{0.1}$
• $F = 500\frac{(1.1)^4 - 1}{0.1}$
• $F = 500\frac{1.4641 - 1}{0.1}$
• $F = 500\frac{0.4641}{0.1}$
• $F = \text{R}2320.50$

Answer: Kobus will have R2320.50 in his account after 4 years.

Worked Example: Monthly Payments Calculation

Question: Ciza deposits R900 at the end of each month for 29 years into an account earning 8.25% interest per annum compounded monthly. Determine her account balance after 29 years.

Solution:

Step 1: Identify the given information

• $F = x\frac{(1 + i)^n - 1}{i}$
• $x = 900$
• $i = \frac{0.0825}{12}$ (monthly interest rate)
• $n = 29 \times 12 = 348$ (total monthly payments)

Step 2: Calculate using the formula

• $F = 900\frac{\left(1 + \frac{0.0825}{12}\right)^{348} - 1}{\frac{0.0825}{12}}$
• $F = \text{R}1,289,665.06$

Answer: Ciza will have R1,289,665.06 after 29 years.

Worked Example: Finding Required Payment Amount

Question: Kosma wants to save R25,000 for a trip to Canada in 2 years. Her savings account earns 10.7% per annum compounded monthly. How much must she save each month?

Solution:

Step 1: Use the payment formula $x = F \times \frac{i}{(1 + i)^n - 1}$

Where:

• $F = 25,000$
• $i = \frac{0.107}{12}$
• $n = 2 \times 12 = 24$

Step 2: Calculate the monthly payment

• $x = 25,000 \times \frac{\frac{0.107}{12}}{\left(1 + \frac{0.107}{12}\right)^{24} - 1}$
• $x = \text{R}938.80$

Answer: Kosma must save R938.80 each month.

Worked Example: Sinking Fund Calculation

Question: Wellington Courier Company buys a truck for R296,000. The truck depreciates at 18% per annum. The company expects new truck prices to increase by 9% annually. They want to replace the truck in 7 years and need to determine monthly sinking fund deposits if the fund earns 13% per annum compounded monthly.

Solution:

Step 1: Calculate the truck's book value after 7 years

• $A = P(1 - i)^n = 296,000(1 - 0.18)^7 = \text{R}73,788.50$

Step 2: Calculate the expected cost of a new truck in 7 years

• $A = P(1 + i)^n = 296,000(1 + 0.09)^7 = \text{R}541,099.58$

Step 3: Determine the sinking fund target

• $F = \text{New truck cost} - \text{Old truck value}$
• $F = \text{R}541,099.58 - \text{R}73,788.50 = \text{R}467,311.08$

Step 4: Calculate required monthly deposits

• $x = F \times \frac{i}{(1 + i)^n - 1}$

• Where $F = 467,311.08$, $i = \frac{0.13}{12}$, $n = 7 \times 12 = 84$

• $x = \text{R}3,438.77$

Answer: The company must deposit R3,438.77 each month into the sinking fund.