Lesson Notes By Weeks and Term v5 - Grade 10

Lesson Video

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Performance objectives

• Calculate simple and compound interest.
• Solve real-life problems involving investments and loans.
• Understand and calculate hire purchase agreements.
• Understand and calculate the effects of inflation.

Lesson summary

This week, we delve into the fascinating world of Finance and Growth, focusing on simple and compound interest, hire purchase agreements, and inflation. Understanding these concepts is crucial for making informed financial decisions throughout your life. Whether you're saving for further education, buying a car, or planning for the future, these mathematical tools will empower you to navigate the financial landscape effectively. Many South Africans struggle with debt and financial literacy; by mastering these concepts, you will be better equipped to make sound financial choices and contribute to a stronger South African economy.

Lesson notes

2.1 Simple Interest Simple interest is a method of calculating interest where interest is earned only on the principal amount. The principal is the initial amount of money borrowed or invested.

Formula: `Simple Interest (SI) = P r * t` Where: `P` = Principal amount `r` = Annual interest rate (expressed as a decimal) `t` = Time period (in years)

Accumulated Amount (A): This is the total amount you will have after the interest is added to the principal. `A = P + SI` `A = P (1 + rt)` Example 1: Sipho invests R5000 in a fixed deposit account that pays simple interest at a rate of 8% per annum. How much interest will he earn after 3 years? What is the total accumulated amount?

Solution: P = R5000 r = 8% = 0.08 t = 3 years SI = P r t = R5000 0.08 * 3 = R1200 A = P + SI = R5000 + R1200 = R6200 Therefore, Sipho will earn R1200 in interest after 3 years, and the total accumulated amount will be R6200. 2.2 Compound Interest Compound interest is calculated on the principal amount and also on the accumulated interest of previous periods. This means you earn interest on your interest!

Formula: `A = P (1 + r/n)^(nt)` Where: `A` = Accumulated amount `P` = Principal amount `r` = Annual interest rate (expressed as a decimal) `n` = Number of times interest is compounded per year `t` = Time period (in years)

Common Compounding Periods: Annually: n = 1 Semi-annually: n = 2 Quarterly: n = 4 Monthly: n = 12 Example 2: Thandi invests R8000 in an account that pays compound interest at a rate of 10% per annum, compounded quarterly. How much will she have after 5 years?

Solution: P = R8000 r = 10% = 0.10 n = 4 (compounded quarterly) t = 5 years A = P (1 + r/n)^(nt) = R8000 (1 + 0.10/4)^(45) = R8000 (1.025)^20 ≈ R13059.84 Therefore, Thandi will have approximately R13059.84 after 5 years. 2.3 Hire Purchase Agreements Hire purchase (HP) is a method of purchasing goods where the buyer makes an initial deposit and pays the remaining balance in installments over a specified period, including interest. The buyer only owns the goods fully after all installments have been paid.

Calculation: Calculate the principal amount borrowed (price of item - deposit). Calculate the total interest payable on the principal amount (using simple interest). Calculate the total amount payable (principal + interest). Calculate the monthly installment (total amount payable / number of months).

Example 3: A fridge costs R

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0. You pay a 15% deposit and finance the rest over 2 years at a simple interest rate of 12% per annum. Calculate the monthly installment.

Solution: Deposit = 15% of R12000 = 0.15 * R12000 = R1800 Principal amount = R12000 - R1800 = R10200 Interest = P r t = R10200 0.12 2 = R2448 Total amount payable = R10200 + R2448 = R12648 Monthly installment = R12648 / (2 * 12) = R12648 / 24 = R527 Therefore, the monthly installment is R527. 2.4 Inflation Inflation is the rate at which the general level of prices for goods and services is rising, and subsequently, purchasing power is falling. A Rand buys less than it did before.

Formula to calculate future price: `Future Price = Present Price (1 + i)^n` Where: `i` = Inflation rate (expressed as a decimal) `n` = Number of years Formula to calculate real value of money: `Real Value = Future Value / (1 + i)^n` Example 4: A loaf of bread currently costs R

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5. If the inflation rate is 6% per year, what will the price of the loaf of bread be in 4 years?

Solution: Present Price = R15 i = 6% = 0.06 n = 4 years Future Price = R15 (1 + 0.06)^4 = R15 * (1.06)^4 ≈ R18.93 Therefore, the loaf of bread will cost approximately R18.93 in 4 years.

Example 5: You will receive R10,000 in 3 years. If the average inflation rate is expected to be 5% per year, what is the real value of the R10,000 in today's money?

Solution: Future Value = R10,000 i = 5% = 0.05 n = 3 years Real Value = R10,000 / (1 + 0.05)^3 = R10,000 / (1.05)^3 ≈ R8638.38 Therefore, the real value of R10,000 in 3 years, in today's money, is approximately R8638.

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8. Guided Practice (With Solutions)

Question 1: Calculate the simple interest earned on an investment of R7500 at an interest rate of 9% per annum for 2.5 years.

Solution: P = R7500 r = 9% = 0.09 t = 2.5 years SI = P r t = R7500 0.09 2.5 = R1687.50 The simple interest earned is R1687.

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0. Question 2: Calculate the accumulated amount if R12000 is invested at a compound interest rate of 7.5% per annum, compounded monthly, for 4 years.

Solution: P = R12000 r = 7.5% = 0.075 n = 12 t = 4 years A = P (1 + r/n)^(nt) = R12000 (1 + 0.075/12)^(124) = R12000 * (1.00625)^48 ≈ R16160.98 The accumulated amount is approximately R16160.

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8. Question 3: A television costs R

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0. You pay a 20% deposit and agree to pay the remainder in monthly installments over 18 months at a simple interest rate of 15% per annum. Calculate the monthly installment.