Simple and Compound Interest

Interest

Interest is the fee that the lender imposes to the borrower in addition to the principal loan amount. Interest is the cost of renting money and the interest rate is the amount of interest payable in proportion to the principal amount. 
Money's value does not stay constant throughout time. With time, it evolves. The value of Rs.100 now won't remain the same tomorrow. If Rs.100 could currently be used to purchase 5 chocolates, Rs.100 might only be able to purchase 4 of the same chocolate in the future. Inflation or a price increase could be to blame.
If there is a greater chance of risk and failure on the borrower's side, a higher interest rate is charged; conversely, if there is a lesser chance of risk and failure on the borrower's side, the interest rate will be reduced. This is why interest rates are so relevant and significant at the same time. They have a significant impact on markets and the economy, and they are a key component of the curriculum in MBA and PGDM programs. 
The interest is added to the principal after every conversion period. The conversion period is the predetermined period of time at the end of which interest is computed and added to the principal at the beginning of the next period. In other terms, the conversion period is the time frame during which interest is compounded.
The conversion period is six months when the interest is computed and added to the principal every six months. Similar to how the interest is calculated and added on a quarterly basis, the conversion period is 3 months.
 If no conversion term is mentioned, one year is assumed to be the conversion period.

Interests can be classified as



Simple Interest

Simple interest is a type of interest which once credited does not earn interest on itself and remains fixed over time.
The formula to calculate Simple Interest is

Where

P = Principal Sum, the initially deposited amount or the original loan

R = rate of interest, the rate at which loan interest is charged

T = Time period, the duration for which money is borrowed or deposited 

So, if an amount of P is borrowed at the rate of interest R for the period of T years, then the amount to be repaid to the lender at the end of conversion period is

 



Some Special Cases of Simple Interest


1. If the rate of interest is r1% for T1 years, r2% for T2 years …. rn% for Tn years of investment and the Simple Interest obtained is Rs. x on this investment, then the principal amount is given by



2. If a person deposits a sum of Rs. X at r1% per annum and the sum of Rs. Y at r2% per annum, then the rate of interest for the total sum deposited is given by

 

3. If a certain amount of money becomes x times in T years, at a specific Simple Interest, then the rate of interest per annum is given by



4. If a certain amount of money is lent out in n parts in such a way that an equal sum of money is available at simple interest on each part at the given interest rates at R1%, R2%, …, Rn% respectively, and periods are mentioned as T1, T2, …, Tn respectively, then the ratio in which the sum that will be divided into n parts is given by




Compound Interest

Interest that is calculated on both the principal and previous interest is known as compound interest. When interest is calculated using both the principal and interest from previous periods, the process is known as compounding. Therefore, interest on principal and interest from the prior period are included in the total interest for the succeeding period. "Interest on interest" is the phrase for it.

It differs from Simple Interest, where there is no compounding because prior interest is not added to the principle for the current month.

This the most common type of interest that is used in the banking system and economics. Here, interest along with one principal further earns interest on it after the completion of every conversion period. Suppose an amount P is deposited in an account or lent to a borrower that pays compound interest at the rate of R% per annum, then after n years the deposit or loan will accumulate to:


The compound interest generated in this period is


In simple representation,






The compound interest can be calculated annually, half yearly, quarterly or monthly.

    If the rate of interest for first, second and third years are R1%, R2% and R3% respectively, then, the amount




    Present worth of Rs. x due n years hence is given by:



    If a certain amount of money becomes x times in n years, then the rate of compound interest during the period will be




    If a sum of money P amounts to A1 after T years at a compound interest, compounded annually and the same sum of money amounts to A2 after (T + 1) years at same compound interest, then the rate of interest is given by



    Applications of Compound Interest Formula 


    The compound interest formula can be used in solving various real-life problems mathematically. Some of the applications of the compound interest formula:

    1. Growth and reduction in the population
    2. Increase and decrease in prices of items
    3. Increase and decay in the amount of microorganisms or radioactive elements
    4. Inflation or deflation in the profit and loss

     

    Solved Examples

    Qno.1: A student purchases a laptop with the help of a low-interest loan from a finance company.  laptop costs Rs. 15000, and the loan charges a 12% interest rate. Calculate the following if the loan is to be paid back in weekly instalments over a period of two years:

    1. The total interest paid during a two-year period,
    2. The whole amount that must be repaid,
    3. The weekly instalment amount. 

    Solution: Given: Principal P = Rs. 15000

    Rate of interest R = 12% simple interest

    Time period T = 2 years

    The simple interest for two years can be calculated as

    =Rs. 3600

    The total amount to be paid after two years =Principal + Interest for two years

    =Rs.15000+Rs.3600

    =Rs.18600

    Now, the weekly instalments amount

    There will be 52 weeks in an year.

    Amount of weekly pay out                                               

    =Rs.178.8 per week

     

    Qno.2: Rachana took a loan of Rs.2450 for 6 years and Rs.3600 for 3 years at the same interest rate. She was paid an interest of Rs.1275 for both. Calculate the interest rate?

    Solution: Since the mode of interest is not specified, we can consider this as simple interest.

    For P=Rs.2450 and T=6 years

    For P=Rs.3600 and T=3 years

    Now,


    ⇒1275=24.5×R×6+36×R×3

    ⇒1275=147R+108R

    ⇒1275=255R

    Hence, the interest will be 5%.


    Qno.3: Akhil purchased a second hand car for Rs.55000 on the terms that he should pay Rs.4275 as cash down payment and the rest in three equal installments. The car dealer charges interest at the rate of 16% per annum compounded half-yearly. Find the value of each installment to be paid by Akhil.

    Solution: The cost of the car is Rs.55000.

    Akhil pays a down payment of Rs.4275.

    So, the remaining amount subjected to loan =Rs.55000-Rs.4275= Rs.50725

    The rate of interest, R = 16% compounded Half-yearly in 3 equal instalments.

    Let x be the amount of installment. Then,



    ⇒Rs.50725 = x (0.79421 + 0.85722 + 0.9259)

    ⇒Rs.50725 = x (2.577)

     

    Hence, Akhil has to pay as installment.