Compound interest means "charging/earning interest on interest. When we borrow some money from any person or bank, then we pay a "fee or cost of borrowing that money" at the time of repaying along with the principle amount. This fee or cost of money is known as interest. When interest is calculated on " principal+ interest earned / charged previously", that interest is known as compound interest.

Example: A man deposited Rs.10,000 in the bank at 8 % per annum for 2 years. Calculate compound interest annually.
Solution: After first year interest = 8 % of 10,000 = 800
and interest added to principal, then total money will be 10,800 and for second year interest calculated on 10,800
interest of second year = 8 % of 10,800
= 864
total interest in two years = 800 + 864
= 1,664

where, A = Amount
P = Principal
R = rate of interest
n = time

Example: Find the compound interest on ₹ 5000 at 5% per annum for 2 years,compounded annually.
Solution: given, P = ₹ 5000
R = 5 %
T = 2 yrs

A = P		1 +	R		n
			100

= 5000		1 +	5		2
			100

= 5000		1 +	1		2
			20

= 5000		21		2
		20

= 5000		21		×		21
		20				20

= 50		21		×		21
		2				2

= 25 ×		441
		2

= 5512.5
∴ A = ₹ 5512.5
Compound Interest = Amount - Principal
= 5512.5 - 5000
= ₹ 512.5

We know that,

= P		1 +	R		n
			100

where, A = Amount
P = Principal
r = rate of interest
n = time
CI = Compound Interest

Example: Find the compound interest on ₹ 8000 in 2 yr at 4 % pa, compounded annually.
Solution: given, P = ₹ 8000
r = 4 %
T = 2 yr

CI = P		1 +	R		n - 1
			100

= 8000		1 +	4		2 - 1
			100

= 8000		1 +	1		2 - 1
			25

= 8000		26		2 - 1
		25

= 8000		676		- 1
		625

= 8000		676 - 625
		625

= 8000		51
		625

= ₹ 652.8

Formula: If interest is compounded half- yearly,

Example: Find the compound interest on ₹ 6000 in 1 yr at 10 % pa, if the interest being compounded half-yearly.
Solution: given, P = ₹ 6000
r = 10 %
n = 1 yr

A = P		1 +	R		2n
			2 × 100

A = P		1 +	R		2 × 1
			2 × 100

= 6000		1 +	10		2
			2 × 100

= 6000		1 +	1		2
			20

= 6000		21		2
		20

= 6000		21		×		21
		20				20

= 15 × 441
= ₹ 6615

Formula - If interest is compounded quarterly,

Example: Find the compound interest on ₹7000 at 8 % pa compounded quarterly for 6 months.
Solution: given, P = ₹7000
r = 8 %
n = 6 months = 1/2 year

= 7000		1 +	8		4 ×1/2
			4 ×100

= 7000		1 +	1		4 ×1/2
			50

= 7000		51		4 ×1/2
		50

= 7000		51		×		51
		50				50

= 7000		14 × 51 × 51
		5

= ₹ 7282.8
CI = Amount - Principal
= 7282.8 - 7000
= ₹ 282.8

Formula: If interest is compounded annually but times is in mixed number

,then

Example: Find the compound interest on ₹ 4000 at 15 % pa for 2 yr 4 months, compounded annually.
Solution: given, P = ₹ 4000
r = 15 %

= 7000		1 +	15		2	×		15
			100					b × 100

= 4000		1 +	3		2	×		1
			20					20

= 4000		23		2	×		21
		20					20

= 4000		23		×		23		×		21
		20				20				20

=		23 × 23 × 21
		20

= ₹ 5554.5‬

= 5554.5‬ - 4000
= ₹ 1554.5

Formula: If rate of interest are r₁%, r₂% and r₃% for 1st, 2nd and 3rd yr respectively, then

Example: Find the amount on ₹ 4000 in 3 yr, if the rate of interest is 3 % for 1st yr, 4 % for the 2nd yr and 5 % for the 3rd yr.
Solution:- given, P = ₹ 4000
r₁ = 3 %
r₂ = 4%
r₃ = 5%

A = P

1 +

r1

×

1 +

r2

×

1 +

r3

100

100

100

= 4000		1 +	3		×		1 +	4		×		1 +	5
			100					100					100

= 4000		103		×		26		×		26
		100				25				20

= 4000		2 × 103 × 26 × 21
		25

= ₹ 4499.04

Formula: Difference between Compound interest and Simple Interest for 2 yrs

Where, P = Principal
r = rate of interest
SI = simple interest

Example: The difference between compound interest and simple interest for 2 yr at 5% per annum is ₹ 15, then find the sum.
Solution: given, difference ( D ) = 15
rate = 5%

Or, 15 = P		5		2
		100

Or, 15 = P		1		2
		20

or, 15 = P		1
		400

or, P = 15 × 400
∴ P = ₹ 6000

Formula: Difference between compound interest and simple interest for 3 yrs

Example: The difference between CI and SI for 3 yr at the rate of 10% per annum is ₹ 155, then find the principal.
Solution: given, Difference = 155
r = 10%

Difference = P		r		2	×		r	+ 3
		100					100

or, 155 = P		10		2	×		10	+ 3
		100					100

or, 155 = P		1		2	×		1	+ 3
		10					10

or, 155 = P		1		×		31
		100				10

or, 155 = P		31
		1000

or,= P		155 × 1000
		31

∴ P = 5 × 1000 = ₹ 5000

Formula: If the population of a town is P and it increases with the rate of r%per annum, then

If population decrease, then

Example: The population of a city increase at the rate of 5% per annum. If its population was 5000 at the end of year 2002, then what will be its population at the end of year 2004? Solution: given, r = 5%
n = 2 yr
P = 4000

= 4000		1 +	5		2
			100

= 4000		21		2
		200

= 4000		21		×		21
		20				20

= 10 × 21 × 21
= 4410

Formula: If the population of a town is P and it increases with the rate of r%per annum, then

If Population decrease, then

Example: The population of a city increase at the rate of 10% per annum. If the present population of the city is 6050000, then what was its population 2 yr ago?
Solution: given, P = 605000
r = 10%
n = 2 yr
Population n yr ago = P / { 1+ ( r/100 )}ⁿ

Population n yrs ago =		p		n
		1 + r/100

=		605000		2
		1 + 10/100

=		605000		2
		1 + 1/10

=		605000		2
		11/10

=		605000
		121/100

= 5000 × 100
= 5,00,000