Compound Interest Concept and Formulas

In this post we will discuss the basic concepts of compound interest and how to calculate different values using the formulas. Apart from the concept we will also discuss the shortcut tricks and techniques which will help you solve certain kind off questions in short duration of time.

We have tried to make the material as comprehensive as possible, if you feel that we have left any topic, please feel free to drop a mail regarding the same.

Compound Interest Concept and Tricks

What is Interest?

Interest is the amount charged on money that is borrowed.

Suppose you borrowed Rs. 100 from bank at 5% rate of interest per annum.
So the interest you have to pay is (5/100) * 100 = Rs. 5 per annum

Types of Interest

Concept of Interest is categorized in two types:
–> Simple Interest
–> Compound Interest

Here we will study only compound interest

What is Compound Interest?

Compound Interest (C.I) is interest calculated on the amount that includes the principal plus accumulated interest. In simple words Interest on Interest.

For example,
Let’s suppose you borrowed Rs. 100 from bank at 5% compound interest for 3 years. Find the compound interest you have to pay at the end of 3 years.

Interest for First year
==> 100 * 5/100 ==> Rs 5

Now this interest will be added back to the principal
=> Principal + Interest
=> 100 + 5
=> Rs. 105

Interest calculation for second year
==> 105 * 5/100==> Rs 5.25
You can see that for second year interest calculation, we have taken different principal which is accumulation of (principal + interest)

Now again principal will be
=> Previous principal + Interest
=> 105 + 5.25
=> Rs. 110.25

Interest calculation for third year
==> 110.25 * 5/100 ==> Rs. 5.51

So total interest payment will be => Interest 01 + Interest 02 + Interest 03
==> Rs 5 + Rs 5.25 + Rs 5.51
==> Rs 15.76

Hence you can see that with compound interest, the interest for each year is different because each year principal gets different.

Also the total amount paid can be written as
Amount = Principal + total interest
Amount = 100 + 15.76 = Rs. 115.76
Hence after three years, you will have to pay total amount of Rs. 115.76 to the bank

Formula for Amount

A=\quad P\quad { [1+\frac { r }{ 100 } }]^{ n }

Where
A => Amount
P => Principal
r => rate of interest
n => time in years

This is the most basic formula for amount calculation if the interest is calculated using compound interest. Please remember this formula as it would help us solving the questions.

Formula for Compound Interest

The formula for compound interest can be written as
Compound Interest => Amount – Principal

Another formula,
CI= \quad P\quad [{ (1+\frac { r }{ 100 } })^{ n }\quad -\quad 1]

Where,
C.I = Compound Interest
P = Principal
r = rate of interest per annum
n= time in years

Compound Interest Calculations

There are different ways of compound interest calculation.
1. Interest compounded annually
2. Interest compounded Quarterly
3. Interest compounded semi-annually

1.Interest Compounded annually

This means that interest is calculated after the end of each year.
So if you borrow money for 3 years, the interest will be calculated three times at the end of each year

Formula used is
A=\quad P\quad { (1+\frac { r }{ 100 } })^{ n }

Example 01
Find the compound interest on Rs. 1200 at 10% per annum for 3 years, compounded annually?

Solution
Principal (P) = Rs. 1200
Rate of interest (r) = 10% per annum
time (n) = 3 years

A=\quad P\quad { (1+\frac { r }{ 100 } })^{ n }\\\ \\ A=\quad 1200\quad (1+\frac { 10 }{ 100 } )^{ 3 }\\\ \\ A=\quad 1200\quad (\frac { 110 }{ 100 } )^{ 3 }\\\ \\ A=\quad Rs.\quad 1597.2

Hence the amount paid at the end of three years is Rs. 1597.2

The total interest paid is
Compound Interest = Amount – Principal
Compound Interest = 1597.2 – 12000 => Rs. 397.2

Hence the total interest paid at the end of years is Rs. 397.2

2. Interest calculated annually but time is given in fraction

Let us suppose that you have to calculate compound interest in the time duration of \ n\frac { x }{ y } years.

Then the formula for computation of amount is
A=\quad P\quad { (1+\frac { r }{ 100 } })^{ n }\quad \times \quad { (1+\frac { \frac { x }{ y } r }{ 100 } })

Example 01
Find compound interest on Rs. 6000 at 6% per annum for 1\frac { 1 }{ 2 } \quad years compounded annually.

Solution
Using the above formula
A=\quad P\quad { (1+\frac { r }{ 100 } })^{ n }\quad \times \quad { (1+\frac { \frac { x }{ y } r }{ 100 } })\\\ \\ \ A=\quad 6000\quad { (1+\frac { 6 }{ 100 } })^{ 1 }\quad \times \quad { (1+\frac { \frac { 1 }{ 2 } \times 6 }{ 100 } })\\\ \\ A=\quad 6000\quad \times \frac { 106 }{ 100 } \times \frac { 103 }{ 100 } \\\ \\ A=\quad Rs.\quad 6550.8\

Hence the total amount paid is Rs 6550.8

Now the total interest paid is
C.I = Amount – Principal
C.I = 6550.8 – 6000
C.I = Rs 550.8

Hence the total interest paid is Rs. 550.8

3. When interest is calculated quarterly

This means that interest is calculated at the end of every three months.
Suppose you borrow Rs. 100 from bank for 1 year compounded quarterly. Here your interest will be calculated 4 times every three months

Formula when interest is compounded quarterly

A=\quad P\quad { (1+\frac { r/4 }{ 100 } })^{ 4n }\

where
A = Amount
P = Principal
r = rate of interest
n = time in year

In formula we have;
==> 4n and r/4 because interest is calculated four times a year

Example 01
Find compound interest on Rs. 16000 at 20% per annum for 9 months, compounded quarterly?

Solution
A=\quad P\quad { (1+\frac { r/4 }{ 100 } })^{ 4n }\

R = 20%
R/4 = 5%

n = 9/12 = 3/4 years
4n = 3

A=\quad 16000\quad { (1+\frac { 5 }{ 100 } })^{ 3 }\\\ \\ A=\quad 16000\quad { (\frac { 21 }{ 20 } })^{ 3 }\\\ \\ A=\quad 18522\

Hence the total amount to be paid after 9 months is Rs. 18522

Interest Paid is = Amount – Principal
Interest = 18522 – 16000
Interest = Rs. 2522

Hence the total interest paid is Rs 2522

4. When interest is calculated half yearly

Interest compounded half yearly means that interest is calculated every six months, i.e. two times in a given year

The formula for amount is given as
A=\quad P\quad { (1+\frac { r/2 }{ 100 } })^{ 2n }

Here we have written r/2 and 2n because the calculation of interest is done twice a year.

Example 01
Find the Compound Interest on Rs. 10,000 at 4% per annum for 2 years compounded half-yearly?

Solution
A=\quad P\quad { (1+\frac { r/2 }{ 100 } })^{ 2n }

Principal = Rs. 10,000
Rate = 4%
R/2 = 2%

n= 2 years
2n = 4 half years

A=\quad 10,000\quad { (1+\frac { 2 }{ 100 } })^{ 4 }\\\ \\ A=\quad 10,000\quad (\frac { 102 }{ 100 } )^{ 4 }\\\ \\ A=\quad 10824.32

The amount paid is Rs 10824.23

The total interest paid = Amount – Principal
Interest = 10824.23 – 10,000
Interest = Rs. 824.23

Hence the total interest paid is Rs. 824.23

5. When Rates are different for different years

Let there are two rates, R1 for year 1 and R2 for year 2. Then the amount can be calculated by using following formula;

A=\quad P\quad { (1+\frac { R1 }{ 100 } })\quad (1+\frac { R2 }{ 100 } )\

Example 01
Rs 2500 was borrowed for three years. What will be the compound interest if the rate of interest for first year is 3% per annum, second year is 4% per annum and for third year is 5% per annum respectively.

Solution
As the rate of interest is different for different year, we have to use following formula

A=\quad P\quad { (1+\frac { R1 }{ 100 } })\quad (1+\frac { R2 }{ 100 } )\quad (1+\frac { R3 }{ 100 } )\quad \

Here
P = Rs 2500
R1 = 3%
R2 = 4%
R3 = 5%

A=\quad P\quad { (1+\frac { 3 }{ 100 } })\quad (1+\frac { 4 }{ 100 } )\quad (1+\frac { 5 }{ 100 } )\\\ \\ A=\quad 2500\quad { (\frac { 103 }{ 100 } })\quad (\frac { 104 }{ 100 } )\quad (\frac { 105 }{ 100 } )\ \quad \

A = Rs 2811.90
Hence the amount to be paid after 3 years is Rs 2811.90

Total Interest Paid = Amount – Principal
Interest = 2811.9 – 2500 => Rs. 311.9

Hence the total interest paid is Rs. 311.9

6. Payment of Loan amount

When you take loan from bank, you pay certain installments to repay the loan. Loan payment type questions are frequently asked in aptitude exams like GRE, GMAT, SSC, CAT etc

In order to solve the questions you need to find the present value of all the installments.
The formula for equal installment is given as:

Each\quad Installment=\quad \frac { P }{ (1+\frac { R }{ 100 } )^{ n } }

Example 01
A builder borrows Rs. 2550 to be paid back with compound interest at the rate of 4% per annum, by the end of 2 years in two equal yearly installments. How much each installment will be?

Given:
Principal = Rs 2550
R = 4% per annum
Time = 2 years
Number of installment = 2

we know that,
Value of Installment = Present value of 1st installment + Present value of 2nd Installment

Each\quad Installment=\quad \frac { 2550 }{ (1+\frac { 4 }{ 100 } )^{ 1 } } \quad +\frac { 2550 }{ (1+\frac { 4 }{ 100 } )^{ 2 } }

On solving the above equation we will get;
Each Installment = Rs. 1352

This is the way you have to solve these types of questions. I strongly suggest to remember the formula as it will help you solve questions fast in the examination.

Shortcut Formulas for Compound interest

(01) If an amount of money grows up to ₹x in ‘n’ years and up to ₹y in ‘n+1’ years on compound interest then the rate of interest can be calculated using the following formula

(02) A sum of money placed at compound interest becomes x times in ‘n’ years and y times in ‘’ years, then two sum relates to

(03) If a certain sum becomes ‘x’ times of itself in ‘n’ years, the rate of compound interest will be equal to