Compound Interest


Introduction

Imagine how beautiful it would be if ice-cream can keep multiplying like you can keep a bowl of ice-cream in the fridge to wake up next morning and find an extra bowl of ice-cream. Wouldn’t it be an amazing treat to gobble down what you love? Similarly, compound interest is a treat for people who work day and night scrupulously to earn money in order to achieve their dreams and ambitions. Compound interest is like multiplication of the money that you keep in your bank account. In fact, earning compound interest is a great thing because it is not just simple interest you get on a certain amount of money. The compound interest is the interest earned on the principal (original amount) as well as on the interest already earned. It also keeps multiplying every year. So let’s delve deeper into the chapter to find out how money exponentially grows every year on the application of compound interest.

S.I and C.I Formula Comparison:

Interest or simple interest is the profit earned on lending a sum of money. It is always calculated on the particular rate of interest for a particular period of time. For example, a person borrows Rs 1000 from a money lender and promises to return the amount in two years. Moneylender asks for a profit of 10% each year which needs to be added with the amount at the end of two years. We know, 10% of 1000 is 100. This means moneylender needs Rs 100 extra as profit each year for lending money. This extra profit earned by the moneylender is called interest. If the interest for one year is Rs100 then for two years it’s Rs 200. So at the end of two years, the person will pay the amount he took from the moneylender along with the simple interest i.e, Rs (1000 + 200 = 1200). Therefore, the simple interest on Rs 1000 at 10% p.a for two years is Rs 200.


In Compound interest class 8, the calculation of compound interest is the same as simple interest every year with the principal (amount on which interest is calculated) renewed each time. If you keep a fixed amount in a bank then every year some interest gets added to it. This interest is not the same but increases every year. For example, to find the compound interest on Rs 1000 for 2 years at 10% per annum, we need to calculate the interest for each year separately. 


Principal for 1st year 

Rs 1000

Interest for 1st year 

10% of 1000 = Rs 100

The amount at the end of 1st year 

Rs 1000+100 = Rs 1100


The amount at the end of the first year will be carried forward to the next year as principal for the second year. So the interest in the second year will be calculated on the new principal which is the sum of the previous year’s principal and earned interest.


Principal for 2nd year

Rs 1100

Interest for 2nd year 

10% of 1100 = Rs 110

The amount at the end of 2nd year 

Rs 1100 + 110 = Rs 1210


Thus, at the end of the second year, the total amount to be paid is Rs 1210 for borrowing Rs 1000. 

Profit earned by moneylender is Rs 1210 - Rs 1000 that is Rs 210. This profit is known as compound interest.


Simple interest and Compound interest for one year is always the same if it is compounded annually. The formula for calculating simple interest as well as compound interest for one year:


S.I for 1yr = C.I for 1 yr 

\[\frac{Principle \times Rate}{100}\]

Terms Related to Compound Interest


Principal - The sum of money lent for a certain period of time at a particular rate of interest.


Time - It is the duration for which the principal is lent, mostly calculated in years.

Interest - It is the profit earned on lending a principal for a certain period of time.

Compound interest- It is the total annual interest earned on lending a principal for a certain period of time


Rate - It is the percentage of interest earned lending a sum of money.


Amount- Amount is the final amount of money left at the end. It is the sum of the original principal and the total compound interest earned.


Calculations for each year:

Compound Interest Formula:

Interest

\[\frac{{Principle \times Rate}}{{100}}\]


Principal

\[\frac{{Interest \times 100}}{{Rate}}\]

Rate

\[\frac{{Interest \times 100}}{{Principal}}\]



Final Amount:


Amount = Principal 

\[(1 + \frac{Rate}{n})^{\text{n time}} \]

[n is the number of times principal is compounded.]

Amount = Principal \[(1 + \frac{Rate}{100})^{\text{time in years}} \] ---- if compounded annually.



Compound Interest formula in Maths:


Compound Interest = Final amount - Original amount.

                                 = Principal 

\[(1 + \frac{Rate}{n})^{n time} \] - Principal



Compound Interest Half Yearly Formula:


If the calculation of compound interest is not annual then the rate of interest also needs to be calculated in accordance. If interest is compounded half yearly then the rate of interest also needs to be divided by 2 is the given rate of interest is for per annum. 


Rate of interest for half year = R / 2

A = P [ 1 + ( {R / 2} / 100 ) ]T,

where ‘T’ is the time period.

Compound Interest Examples:


Some of the important Compound Interest Problems are given below:


 1)  Find CI on Rs 15,000 for 2 years at 10% per annum compounded annually.


Solution

Principal for first year                                  Rs 15,000

Interest for first year                                    Rs 1,500 (pxr/100)

Amount at the end of first year                    Rs 16,500 (pxr/100)


Principal for the second year                       Rs 16,500

Interest for the second year                         Rs 1,650

The amount at the end of the second year  Rs 18,150


According to SI and CI formula:


C.I = Final amount - Original amount

    = Rs 18,150 - 15,000

    = Rs 3150. 


Alternative method:

Principal = Rs 15,000

Rate of interest = 10%

n = 2 years


A = 15000 \[(1 + \frac{11}{10})^2 \]

   = 15000 \[\frac{21}{10})^2 \]

   = 15000 \[(\frac{441}{100})\]

   = 18150


Amount = Rs 18,150

C.I = Amount - Principal

      = Rs 18,150 - Rs 15000

      = Rs 3,150


2) What amount is to be repaid on a loan of Rs 20,000 for 1 and a half years at 10% per annum compounded half-yearly.


Solution:

In case of interest compounded half-yearly, we consider a new principal at the end of every six months and calculate interest every six months. 

In the formula, S.I \[ = \frac{P \times T \times R}{100} \]

, time will be ½ year.


Principal for six months     Rs 20,000

Interest for six months       Rs 1,000       (20000x10/200)

Amount after six months    Rs 21,000


Principal for the 2nd six months Rs 21,000

Interest for the 2nd six months    Rs 1,050

Amount after 2nd six months       Rs 22,050


Principal for 3rd six months     Rs 22,050.00

Interest for 3rd six months       Rs 1,102.50

Amount after 3rd six months    Rs 23,152.50


Final Amount = Rs 23,152.50

Principal = 20,000

Compound Interest = Rs 3,152.50


Alternative:

In 1 ½ year, there are three 6months so n= 3.

Since the interest is 10% per annum we need to find the interest for six months 

The rate for 6months = 5%.


Amount

\[ = 20,000(1 + \frac{5}{100})^3 \]

\[ = 20,000(1 + \frac{1}{20})^3 \]

\[ = 20,000(\frac{21}{20})^3 \] 

\[ = 20,000(\frac{21}{20} \times \frac{21}{20} \times \frac{21}{20}) \] 

             = 23,152.50


Compound Interest = Amount - Principal

                                = 23,152.50 - 20,000

                                = 3,152.50


Therefore, Compound Interest is Rs 3,152.50.