Simple and Compound Interest | Math of Finance || Mathematics in the Modern World

The video introduces the concepts of simple and compound interest, explaining why people borrow money and lenders charge interest. It uses an example of Martina borrowing 5000 pesos at 10% interest for one year to illustrate these ideas.

Definitions of essential terms are provided: the debtor (Martina) borrows money, the lender (Tessie) loans money, interest is the payment for borrowing money (500 pesos), the principal is the initial amount (5000 pesos), the rate of interest is the percentage charged (10%), and time is the duration of the loan (1 year). The final amount or maturity value is the sum of the principal and interest (5500 pesos).

Simple interest is calculated only on the original principal using the formula I = PRT (Interest = Principal × Rate × Time). The final amount (F) is P + I, or alternatively F = P(1 + RT). Time (T) must always be in years; if given in months, divide by 12.

An example demonstrates calculating the simple interest earned by investing 200,000 pesos at 5.4% for three years, illustrating how to convert percentages to decimal form for calculation.

Another example shows how to find both the interest and the final amount for 21,000 pesos at 7.25% simple interest for nine months, emphasizing the conversion of months to years.

This section explains how to derive and use the formula for finding the interest rate (R = I / PT) when the principal, interest, and time are known. It includes converting time from years and months into a decimal year format.

An example illustrates how to determine the time period (T = I / PR) required for a principal amount to earn a specific interest at a given simple interest rate.

Compound interest is defined as interest earned on both the original principal and accumulated interest. Key terms like conversion period (m) and total number of conversion periods (n = T × m) are introduced, along with different compounding frequencies (annually, semi-annually, quarterly, monthly).

The formula for the final amount (F) in compound interest is F = P(1 + i)^n, where i is the periodic rate (R/m). The compound interest (I) is then F - P.

An example demonstrates calculating the maturity value of 80,000 pesos at 6% interest compounded semi-annually for three years, detailing the steps to use the compound interest formula.

This example calculates the total amount a borrower paid after two years for a loan of 150,000 pesos at 9% interest compounded annually, highlighting the identification of principal, rate, time, and compounding frequency.

An illustration shows how to find the compound interest for 5500 pesos invested at 8% compounded quarterly for five years and six months, requiring the calculation of the final amount first, then subtracting the principal.

Accumulation is defined as finding the final amount of a principal at a specified time. An example calculates the accumulated amount for 10,400 pesos for two years at 7% compounded monthly.

The video introduces a formula for finding the time period (T) in compound interest using logarithms: T = log(F/P) / (m × log(1 + R/m)). An example demonstrates how long it takes for 5000 pesos to amount to 6100 at 6% compounded quarterly.

A formula is provided for finding the interest rate (R) in compound interest: R = m * ((F/P)^(1/n) - 1). An example calculates the interest rate if 3050 accumulates to 5000 in four years, compounded monthly.