Lesson Notes By Weeks and Term v5 - Grade 12

Lesson Video

This page supports the lesson note with a companion video and a short classroom-ready summary.

For class groups and homework, share this lesson page so learners also get the summary, objectives, and full lesson context.

Performance objectives

• Calculate future and present value using compound interest.
• Calculate future and present value of annuities.
• Apply depreciation methods to calculate the book value of assets.
• Solve financial problems involving sinking funds.

Lesson summary

Finance, growth, and decay are fundamental concepts in mathematics that play a crucial role in understanding various real-world scenarios, especially in the context of personal finance, investment, and economic planning. As South African learners, understanding these concepts will empower you to make informed decisions about your finances, investments, and future economic prospects. This topic explores the mathematical models that describe how money grows through interest accumulation, how assets depreciate in value over time, and how population growth can be predicted.

Lesson notes

2.1 Compound Interest Compound interest is interest calculated on the initial principal, which also includes all of the accumulated interest from previous periods. This means you earn interest on your interest. It's a powerful tool for wealth creation, but also means debts can grow quickly if not managed properly.

Formula: A = P(1 + i)^n Where: A = Accumulated amount (future value) P = Principal amount (initial investment or loan) i = Interest rate per compounding period (expressed as a decimal) n = Number of compounding periods Important Considerations for 'i' and 'n': Annual Compounding: If interest is compounded annually, 'i' is the annual interest rate and 'n' is the number of years.

Monthly Compounding: If interest is compounded monthly, 'i' is the annual interest rate divided by 12, and 'n' is the number of years multiplied by

1

2. Quarterly Compounding: If interest is compounded quarterly, 'i' is the annual interest rate divided by 4, and 'n' is the number of years multiplied by

4. Example 1: Investment Growth Sipho invests R5,000 in a fixed deposit account that pays 8% interest per year, compounded annually. How much will he have after 5 years? P = R5,000 i = 0.08 n = 5 A = 5000(1 + 0.08)^5 A = 5000(1.08)^5 A = 5000 * 1.469328 A = R7,346.64 Sipho will have R7,346.64 after 5 years.

Example 2: Loan Repayment Thandi takes out a loan of R20,000 to start a small business. The interest rate is 12% per year, compounded monthly. How much will she owe after 3 years if she makes no payments? P = R20,000 i = 0.12 / 12 = 0.01 n = 3 12 = 36 A = 20000(1 + 0.01)^36 A = 20000(1.01)^36 A = 20000 * 1.430769 A = R28,615.38 Thandi will owe R28,615.38 after 3 years. 2.2 Present Value Present value (PV) is the current value of a future sum of money or stream of cash flows given a specified rate of return. Discounting is used to find the present value.

It essentially asks the question: how much money would I need to invest today at a certain interest rate to have a specific amount in the future?

Formula: P = A / (1 + i)^n Where: P = Present value (principal) A = Accumulated amount (future value) i = Interest rate per compounding period n = Number of compounding periods Example 3: Calculating Present Value How much would you need to invest today at an interest rate of 10% per year, compounded annually, to have R10,000 in 4 years? A = R10,000 i = 0.10 n = 4 P = 10000 / (1 + 0.10)^4 P = 10000 / (1.10)^4 P = 10000 / 1.4641 P = R6,830.13 You would need to invest R6,830.13 today. 2.3 Annuities An annuity is a series of equal payments made at regular intervals.

There are two main types: Ordinary Annuity: Payments are made at the end of each period.

Annuity Due: Payments are made at the beginning of each period.

Future Value of an Ordinary Annuity: FV = PMT [((1 + i)^n - 1) / i] Where: FV = Future Value PMT = Payment amount per period i = Interest rate per period n = Number of periods Future Value of an Annuity Due: FV = PMT [((1 + i)^n - 1) / i] * (1 + i)

Present Value of an Ordinary Annuity: PV = PMT [(1 - (1 + i)^-n) / i] Present Value of an Annuity Due: PV = PMT [(1 - (1 + i)^-n) / i] * (1 + i)

Example 4: Future Value of an Ordinary Annuity Zola invests R500 per month into a retirement annuity that earns 9% interest per year, compounded monthly. How much will he have after 20 years? PMT = R500 i = 0.09 / 12 = 0.0075 n = 20 12 = 240 FV = 500 * [((1 + 0.0075)^240 - 1) / 0.0075] FV = 500 * [(6.0378 - 1) / 0.0075] FV = 500 * (5.0378 / 0.0075) FV = 500 * 671.706 FV = R335,853.00 Zola will have R335,853.00 after 20 years.

Example 5: Present Value of an Ordinary Annuity You want to receive R2,000 per month for 10 years. How much do you need to invest today at an interest rate of 7% per year, compounded monthly? PMT = R2,000 i = 0.07 / 12 = 0.005833 n = 10 12 = 120 PV = 2000 * [(1 - (1 + 0.005833)^-120) / 0.005833] PV = 2000 * [(1 - (0.4971)) / 0.005833] PV = 2000 * [0.5029/ 0.005833] PV = 2000 * 86.221 PV = R172,442.00 You need to invest R172,442.00 today. 2.4 Depreciation Depreciation is the decrease in the value of an asset over time due to wear and tear, obsolescence, or other factors.

Two common methods are: Straight-Line Depreciation: The asset depreciates by the same amount each year.

Reducing Balance Depreciation: The asset depreciates by a fixed percentage of its remaining book value each year.

Straight-Line Depreciation Formula: Depreciation per year = (Cost - Salvage Value) / Useful Life Book Value = Cost - Accumulated Depreciation Reducing Balance Depreciation Formula: Book Value at end of year = Original Cost (1 - i)^n Where: i = depreciation rate (as a decimal) n = number of years Example 6: Straight-Line Depreciation A company buys a delivery vehicle for R150,

0

0

0. It has a salvage value of R30,000 and a useful life of 5 years. Calculate the annual depreciation and the book value after 3 years.