Compound Interest

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

 Compound Interest = Amount - Principal

 Amount = Principal + Compound Interest

 A = P 1 + R n 100

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,

 Compound Interest = Amount - Principal

 CI = A - P

 = P 1 + R n 100

 ∴ CI = P 1 + R n - 1 100

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,

 then n = 2n ( when time is given in years)

 Rate = half = r (when rate is given yearly) 2%

 A = P 1 + R 2n 2 × 100

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,

 then, n = 4n ( when time is given in years)

 Rate = R % (when rate is given yearly) 4

 A = P 1 + R 4n 4 × 100

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

 suppose time = na/b

,then

 A = P 1 + R n × 1 + ar 100 2 × 100

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

 n = 2 and = a = 4 = 1 b 12 3

 A = P 1 + R n × ab 100 b × 100

 = 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

 CI = Amount - Principal

= 5554.5‬ - 4000
= ₹ 1554.5

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

 A = P 1 + r1 × 1 + r2 × 1 + r3 100 100 100

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

 Difference = P r 2 = SI × R 100 200

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%

 Difference = P r 2 100

 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

 Difference = P r 2 × r + 3 100 100

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

 Population after n yr = P 1 + r 2 100

If population decrease, then

 Population after n yr = P 1 - r 2 100

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

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

If Population decrease, then

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

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