Lesson Notes By Weeks and Term v3 - Senior Secondary 3
Arithematic of finance
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Arithematic of finance
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Subject: General Mathematics
Class: Senior Secondary 3
Term: 1st Term
Week: 2
Theme: Number And Numeration
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Calculate simpleinterest given the principle, rate and time. calculate compound in terest.using the for mula. Determine the depreciation value of an item. Compute the annuities of a givenproblem Compute the amortization in agiven problem. Solve furtherproblems in capitalmarket usinglogarithm table.
Simple interest is calculated only on the principal amount, or on that portion of the principal amount which remains unpaid. It does not compound, meaning interest is not earned on previously accumulated interest.
Principal (P): The initial amount of money borrowed or invested.
Rate (R): The percentage at which the interest is charged or earned per annum (per year). Usually expressed as a percentage, but used as a decimal in calculations (R/100).
Time (T): The duration for which the money is borrowed or invested, usually in years. If given in months, convert to years by dividing by 12 (e.g., 6 months = 6/12 = 0.5 years).
Simple Interest (I): The amount of interest earned or paid.
Amount (A): The total sum at the end of the period, which is the principal plus the interest (A = P + I).
Formula for Simple Interest: $I = \frac{P \times R \times T}{100}$ Formula for Total Amount: $A = P + I \quad \text{or} \quad A = P \left(1 + \frac{RT}{100}\right)$ Worked Example 1.1 (Simple Interest): Mr. Okoro borrowed N50,000 from a microfinance bank at a simple interest rate of 10% per annum for 3 years. (a) Calculate the simple interest. (b) Calculate the total amount Mr. Okoro will repay.
Solution: Given: P = N50,000, R = 10%, T = 3 years. (a)
Simple Interest (I): $I = \frac{P \times R \times T}{100}$ $I = \frac{50,000 \times 10 \times 3}{100}$ $I = \frac{1,500,000}{100}$ $I = N15,000$ (b)
Total Amount (A): $A = P + I$ $A = 50,000 + 15,000$ $A = N65,000$ Compound interest is the interest calculated on the initial principal and also on the accumulated interest of previous periods. It results in a faster growth of money compared to simple interest. The interest can be compounded annually, semi-annually, quarterly, or even monthly.
Principal (P): The initial amount.
Rate (R): The nominal annual interest rate.
Time (T): The total number of compounding periods.
Amount (A): The total sum at the end of the period, including principal and accumulated interest. Formula for Compound Interest (compounded annually): $A = P \left(1 + \frac{R}{100}\right)^T$ If compounded `n` times per year (e.g., semi-annually n=2, quarterly n=4, monthly n=12): $A = P \left(1 + \frac{R/n}{100}\right)^{nT}$ Where `R` is the annual rate, `T` is the number of years. Compound Interest (CI) = A - P Worked Example 1.2 (Compound Interest): Aisha invests N120,000 in a fixed deposit account that offers a compound interest rate of 8% per annum. (a) Calculate the amount in her account after 2 years. (b) Calculate the total compound interest earned.
Solution: Given: P = N120,000, R = 8%, T = 2 years. Compounded annually. (a)
Amount (A) after 2 years: $A = P \left(1 + \frac{R}{100}\right)^T$ $A = 120,000 \left(1 + \frac{8}{100}\right)^2$ $A = 120,000 (1 + 0.08)^2$ $A = 120,000 (1.08)^2$ $A = 120,000 \times 1.1664$ $A = N139,968$ (b)
Compound Interest (CI) earned: $CI = A - P$ $CI = 139,968 - 120,000$ $CI = N19,968$ Depreciation is the reduction in the value of an asset over time due to wear and tear, obsolescence, or usage. There are various methods, but SS3 typically focuses on two: Straight-Line Method (Uniform Depreciation): The asset loses a fixed amount of value each year. Annual Depreciation = (Original Cost - Salvage Value) / Useful Life Book Value after `n` years = Original Cost - (n Annual Depreciation)
Note: This is simpler and may not align with the "using the formula" implied for compound interest, but it's a common method.* Reducing Balance Method (Compound Depreciation): The asset depreciates by a fixed percentage of its current value each year. This method is mathematically analogous to compound interest, but with a decrease instead of an increase.
Original Value / Cost (P): Initial value of the item.
Rate of Depreciation (R): The percentage decrease in value per period.
Number of Periods (n): The number of years or periods. Value after `n` periods ($V_n$): The depreciated value. Formula for Value after `n` periods (Reducing Balance Method): $V_n = P \left(1 - \frac{R}{100}\right)^n$ Total Depreciation = P - $V_n$ Worked Example 1.3 (Depreciation - Reducing Balance): A brand new Keke NAPEP (tricycle) costs N1,200,
0
0
0. Its value depreciates at a rate of 15% per annum on the reducing balance. (a) Calculate its value after 3 years. (b) Calculate the total depreciation over the 3 years.
Solution: Given: P = N1,200,000, R = 15%, n = 3 years. (a) Value ($V_3$) after 3 years: $V_n = P \left(1 - \frac{R}{100}\right)^n$ $V_3 = 1,200,000 \left(1 - \frac{15}{100}\right)^3$ $V_3 = 1,200,000 (1 - 0.15)^3$ $V_3 = 1,200,000 (0.85)^3$ $V_3 = 1,200,000 \times 0.614125$ $V_3 = N736,950$ (b)
Total Depreciation: Total Depreciation = P - $V_3$ Total Depreciation = 1,200,000 - 736,950 Total Depreciation = N463,050 Annuity refers to a series of equal payments or receipts made at regular intervals. Common examples include loan repayments, insurance premiums, pension fund contributions, or regular savings.
Payment (Pmt): The amount of each equal payment.
Interest Rate (r): The interest rate per period (R/100 if R is annual percentage, then divide by compounding frequency if necessary).
Number of Periods (n): The total number of payments/periods.
There are two main types: Ordinary Annuity: Payments are made at the end of each period. (Most common in SS3).
Annuity Due: Payments are made at the beginning of each period. (More complex, typically beyond SS3).
Focus for SS3: Future Value of an Ordinary Annuity This calculates the total accumulated amount of the series of payments at a future point in time, including all payments and the interest earned on them. Formula for Future Value of an Ordinary Annuity (FV): $FV = Pmt \times \frac{((1 + r)^n - 1)}{r}$ Where: $FV$ = Future Value of the annuity $Pmt$ = Periodic payment amount $r$ = Interest rate per period (as a decimal, e.g., 5% = 0.05) $n$ = Total number of periods Worked Example 1.4 (Future Value of an Annuity): Emeka decides to save N5,000 at the end of each year in an investment scheme that pays 6% interest compounded annually. How much will he have accumulated after 5 years?
Solution: Given: Pmt = N5,000, r = 6% = 0.06, n = 5 years. $FV = Pmt \times \frac{((1 + r)^n - 1)}{r}$ $FV = 5,000 \times \frac{((1 + 0.06)^5 - 1)}{0.06}$ $FV = 5,000 \times \frac{((1.06)^5 - 1)}{0.06}$ $FV = 5,000 \times \frac{(1.338225577 - 1)}{0.06}$ $FV = 5,000 \times \frac{0.338225577}{0.06}$ $FV = 5,000 \times 5.63709295$ $FV \approx N28,185.46$
This topic is highly practical and can be easily integrated into various Nigerian contexts.
Personal Finance and Savings: Application: Students can apply simple and compound interest calculations to understand the growth of their savings in bank accounts, cooperative societies (Esusu), or investment schemes. They learn the benefit of starting to save early due to compounding.
Local Context: Discuss the difference between saving with traditional Esusu groups (often simple interest or no interest) versus formal banks (compound interest), and how this affects returns. This helps them evaluate various savings options available locally.
Loans and Debt Management: Application: Amortization helps students understand how loans (e.g., student loans, car loans, business loans from microfinance institutions) are repaid. They can calculate monthly or annual payments and the total cost of borrowing.
Local Context: Analyzing loan offers from local lenders or commercial banks, comparing interest rates and repayment structures. Understanding the implications of high-interest rates common with informal lenders. This prepares them for responsible borrowing.
Business and Investment Decisions: Application: Depreciation is crucial for businesses to assess the value of their assets (e.g., vehicles, machinery, buildings) for financial reporting and planning. Annuities help in evaluating long-term investments or pension plans. Bonds and debentures introduce them to ways companies and governments raise money.
Local Context: A small business owner calculating the depreciated value of his okada (motorcycle taxi) or generator for resale. An investor considering buying Nigerian Treasury Bills or corporate bonds. Understanding income tax and VAT is vital for anyone engaging in business or employment.