Future Value (HSC SSCE Mathematics Standard): Revision Notes

Worked Example 1a: Blake's Investment

Blake invests $7000 over 5 years at a compound interest rate of 4.5% per annum. Calculate the future value after 5 years, correct to the nearest cent.

Solution:

Step 1: Write the future value formula.

$FV = PV(1 + r)^n$

Step 2: Identify the values from the question.

• $PV = 7000$ (Blake's initial investment)
• $r = 0.045$ (convert 4.5% to decimal by dividing by 100)
• $n = 5$ (5 years of annual compounding)

Step 3: Substitute these values into the formula.

$FV = 7000(1 + 0.045)^5$

Step 4: Calculate step by step.

• First calculate: $1 + 0.045 = 1.045$
• Then calculate: $(1.045)^5 = 1.246181937...$
• Finally: $7000 \times 1.246181937 = 8723.273557...$

Step 5: Round to the nearest cent.

$FV = \$8723.27$

Answer: The future value of Blake's investment after 5 years is $8723.27.

What this means: Blake's $7000 investment grew by $1723.27 over the 5-year period due to compound interest.

Worked Example 1b: Finding Present Value with Monthly Compounding

Calculate the present value of an investment that has a future value of $500,000 over 8 years with an interest rate of 8.5% per annum compounded monthly.

Solution:

Step 1: Write the present value formula.

$PV = \frac{FV}{(1 + r)^n}$

Step 2: Identify the values, being careful with monthly compounding.

• $FV = 500000$ (the target amount)
• $r = \frac{0.085}{12}$ (annual rate divided by 12 months)
• $n = 8 \times 12 = 96$ (8 years multiplied by 12 months)

Why we adjust for monthly compounding: When interest compounds monthly, we need to divide the annual interest rate by 12 to get the monthly rate, and multiply the number of years by 12 to get the total number of monthly compounding periods.

Step 3: Substitute the values into the formula.

$PV = \frac{500000}{(1 + \frac{0.085}{12})^{96}}$

Step 4: Calculate step by step.

• First calculate the monthly rate: $\frac{0.085}{12} = 0.0070833...$
• Then: $1 + 0.0070833 = 1.0070833...$
• Then: $(1.0070833)^{96} = 1.96913...$
• Finally: $\frac{500000}{1.96913} = 253916.41$

Step 5: Round to the nearest cent.

$PV = \$253,916.41$

Answer: The present value is $253,916.41.

What this means: If you want to have $500,000 in 8 years with monthly compounding at 8.5% per annum, you need to invest $253,916.41 today.

Worked Example 2: Tyson's Investment and Interest Earned

Tyson invests $10,000 over 10 years at a compound interest rate of 7½% per annum. Calculate: a) The amount of the investment after 10 years, correct to the nearest cent b) The interest earned after 10 years, correct to the nearest cent

Solution:

Part a) Finding the future value:

Step 1: Write the future value formula.

$FV = PV(1 + r)^n$

Step 2: Identify the values.

• $PV = 10000$ (Tyson's initial investment)
• $r = 0.075$ (convert 7½% or 7.5% to decimal)
• $n = 10$ (10 years of annual compounding)

Step 3: Substitute the values.

$FV = 10000(1 + 0.075)^{10}$

Step 4: Calculate.

• $(1.075)^{10} = 2.061031...$
• $10000 \times 2.061031 = 20610.3156...$

Step 5: Round to the nearest cent.

$FV = \$20,610.32$

Answer for part a: The amount of the investment after 10 years is $20,610.32.

Part b) Finding the compound interest:

Step 1: Write the interest formula.

$I = FV - PV$

Step 2: Substitute the values we know.

• $FV = 20610.32$ (from part a)
• $PV = 10000$ (original investment)

$I = 20610.32 - 10000$

Step 3: Calculate.

$I = \$10,610.32$

Answer for part b: The interest earned after 10 years is $10,610.32.

What this means: Tyson's investment more than doubled over the 10-year period. The compound interest effect meant he earned $10,610.32, which is more than his original investment of $10,000.