Lesson Notes By Weeks and Term v5 - Grade 12
Finance, growth and decay – Week 8 focus
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Finance, growth and decay – Week 8 focus
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Subject: Mathematics
Class: Grade 12
Term: 1st Term
Week: 8
Theme: General lesson support
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Finance, growth, and decay are fundamental mathematical concepts that directly impact your lives and financial futures. This week, we'll delve deeper into compound interest, annuities, and depreciation, equipping you with the skills to make informed financial decisions. Understanding these concepts is crucial for saving money, taking out loans, and planning for your future, especially in the South African context where economic fluctuations and personal financial planning are essential. Think about opening a savings account, taking out a student loan, or buying a car - all of these involve the principles we will cover.
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 that you earn interest on your interest, leading to exponential growth over time.
The formula for compound interest is: A = P(1 + i)^n Where: A = Future Value (the amount you'll have after n periods) P = Principal (the initial amount of money) i = Interest rate per period (annual interest rate divided by the number of compounding periods per year) n = Number of periods (number of years multiplied by the number of compounding periods per year)
Example 1: Thando invests R5,000 in a savings account that pays 8% interest per year, compounded quarterly. How much will she have after 5 years? P = R5,000 i = 8%/4 = 0.08/4 = 0.02 (quarterly interest rate) n = 5 years 4 quarters/year = 20 quarters A = 5000(1 + 0.02)^20 A = 5000(1.02)^20 A = 5000(1.485947) A = R7,429.74 Therefore, Thando will have R7,429.74 after 5 years. 2.2 Annuities: An annuity is a series of equal payments made at regular intervals.
There are two main types of annuities: Future Value Annuity: Used to calculate the total value of a series of payments made over time, accumulating interest. This is common for savings goals like retirement.
Present Value Annuity: Used to calculate the present-day value of a series of future payments. This is common for loans and mortgages.
Future Value Annuity Formula: FV = PMT * [((1 + i)^n - 1) / i] Where: FV = Future Value of the annuity PMT = Payment amount per period i = Interest rate per period n = Number of periods Present Value Annuity Formula: PV = PMT * [(1 - (1 + i)^-n) / i] Where: PV = Present Value of the annuity PMT = Payment amount per period i = Interest rate per period n = Number of periods Example 2: Sipho wants to save R100,000 for a down payment on a house in 5 years. He can deposit R1,500 at the end of each month into an account that pays 6% interest compounded monthly. Will he reach his goal? PMT = R1,500 i = 6%/12 = 0.06/12 = 0.005 (monthly interest rate) n = 5 years 12 months/year = 60 months FV = 1500 * [((1 + 0.005)^60 - 1) / 0.005] FV = 1500 * [((1.005)^60 - 1) / 0.005] FV = 1500 * [(1.34885 - 1) / 0.005] FV = 1500 * [0.34885 / 0.005] FV = 1500 * 69.77 FV = R104,655 Yes, Sipho will reach his goal. He will have R104,655 after 5 years, exceeding his target of R100,
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0. Example 3: Zanele takes out a loan of R50,000 to start a small business. The loan has an interest rate of 12% per year, compounded monthly, and she will repay it over 3 years. What is her monthly payment? PV = R50,000 i = 12%/12 = 0.01 (monthly interest rate) n = 3 years 12 months/year = 36 months PMT = ? Rearranging the Present Value Annuity formula: PMT = PV * [i / (1 - (1 + i)^-n)] PMT = 50000 * [0.01 / (1 - (1.01)^-36)] PMT = 50000 * [0.01 / (1 - 0.698925)] PMT = 50000 * [0.01 / 0.301075] PMT = 50000 * 0.033214 PMT = R1,660.70 Zanele's monthly payment is R1,660.70. 2.3 Depreciation: Depreciation is the decrease in the value of an asset over time due to wear and tear, obsolescence, or other factors.
We will cover two common methods: Straight-Line Depreciation: The asset depreciates by the same amount each year. Depreciation Amount = (Cost - Salvage Value) / Useful Life Book Value = Cost - Accumulated Depreciation Reducing Balance Depreciation: The asset depreciates by a fixed percentage of its book value each year. This results in higher depreciation expenses in the early years and lower expenses later on. Depreciation Rate = 1 - (Salvage Value / Cost)^(1 / Useful Life) Depreciation Amount = Book Value * Depreciation Rate New Book Value = Previous Book Value - Depreciation Amount Example 4: A delivery van costs R200,
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0. It has an estimated useful life of 5 years and a salvage value of R40,
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0. Calculate the depreciation expense for each year using both the straight-line and reducing balance methods.
Straight-Line: Depreciation Amount = (200000 - 40000) / 5 = R32,000 per year Year | Book Value at Start | Depreciation | Book Value at End ------- | -------- | -------- | -------- 1 | R200,000 | R32,000 | R168,000 2 | R168,000 | R32,000 | R136,000 3 | R136,000 | R32,000 | R104,000 4 | R104,000 | R32,000 | R72,000 5 | R72,000 | R32,000 | R40,000 Reducing Balance: Depreciation Rate = 1 - (40000 / 200000)^(1 / 5) = 1 - (0.2)^(0.2) = 1 - 0.67205 = 0.32795 or 32.8% Year | Book Value at Start | Depreciation | Book Value at End ------- | -------- | -------- | -------- 1 | R200,000 | R65,590 | R134,410 2 | R134,410 | R44,078.74 | R90,331.26 3 | R90,331.26 | R29,624.42 | R60,706.84 4 | R60,706.84 | R19,909.11 | R40,797.73 5 | R40,797.73 | R797.73| R40,000 Guided Practice (With Solutions)
Question 1: Sarah invests R10,000 in a fixed deposit account that pays 7% interest per year, compounded monthly. How much will she have after 3 years?