Compound Interest
Compound Interest :
The amount after first unit of time becomes the principal for second unit of time the amount after second unit of time becomes the principal for third unit of time and so on .
After a specified period , the difference between the amount and the principal is called Compound Interest abbreviated as C.I.
Sometime C.I. fix up a certain unit of time, say yearly or half yearly or quarterly to settle the previous account .
Formula :
Let Principal = P,
Time = n,
Rate = R% per annum .
1) When Interest is compounded Annually :
Amount = P [ 1 + R/ 100 ]^n
2) When Interest is compounded Half - yearly :
Amount = P [ 1 +( R/2 )/100]^2n
3) When Interest is Compounded Quarterly :
Amount = p [ 1 +( R/4 )/100 ]^4n
4) When Time is Fraction of year :
3
Le the time be a fraction of a year, say 2------ years
7
Then, Amount = P [ 1 + R/100 ]^2 X [ 1 + ( 3R/7 )/100 ]
5)When Rates are different for different years , say R1, R2, R3, percent for Ist, "nd and £rd year respectively .
Then, Amount =P [ 1 + R1/100 ][ 1 + R2/100 ][ 1 + R3/100 ]
1) Rule :
If Principal P , Rate R% and Time t then
Amount = P [1+ R/100 ]^t
Example :
Find Amount on Rs 6250 at 14% per annum for 2 years, compounded annually.
Sol.
A = 6250[1+14/100 ]^2
= 8122.50
2) Rule :
If Principal P , Rate R% and Time t then
Compound Interest = P [(1+ R/100 )^t -1 ]
Example :
Find Compound Interest on Rs 1000 at 5% per annum for 2 years, compounded annually .
Sol.
Compound Interest = 1000 [( 1+ 5/100 )^2 -1]
= 102.50
3)Rule :
When Rates are different for different years , say R1, R2, R3, percent for Ist, "nd and £rd year respectively .
Then, Amount =P [ 1 + R1/100 ][ 1 + R2/100 ][ 1 + R3/100 ]
Example:
Find Amount on Rs 500 at 10% per annum for 2 years, 10% per annum for 2 years and 5% per annum for 1 year compounded annually.
Sol.
Amount = 500 (1+10/100)^2 (1+10/100)^2 (1+ 5/100)^1
= 500 X 11/10 X 11/10 X 11/10 X 11/10 X 21/20
= 11 X 11 X 11 X 11 X 21 / 400
= 307461 / 400
= 768.652 Rs
4) Rule :
Difference between simple interest an compound interest
1) Difference for two years
Difference = P (Rate/100)^2
P = Difference X 100 X 100/(Rate)^2
2)Difference for three years
Difference = P ( Rate/100)^2 (3+Rate/100)
Example:
Find the difference between compound interest and simple interest for Rs 1000 at the rate 5% for 3 years
Sol.
Difference =1000 (5/100)^2 (3+5/100)
=1000 X 21 X 21 / 20 X 20 X 61/20
=26901 / 8
=3362.62 Rs
5) Rule :
Difference between compound interest and simple interest at rate x% per annum for 2 years
= ( x ^2 /100 )%
Example :
Difference of 5% = 25/100 = 1/4 = 0.25 %
Difference of 7% = 49/100 = 0.49 %
Difference of 8% = 64/100 = .64%
Difference of 10% = 100/100 = 1%
6) Rule:
If any quantity at the rate of compound interest in t years will be x and in 2t years will be y then principal = x^2/y
Example :
If any quantity in 5 years will be Rs 500 and 10 years will be 1000 . What will be the principal .
Sol.
principal = (500 X 500)/1000
=250 Rs
Exercise
1)The difference between compound interest and simple interest on Rs 1250 for 2 years at 8% is:
a) Rs 2 b) Rs 4 c) Rs 6 d) Rs 8
2)The compound interest on Rs .2000 for 3 years is Rs . 315 . 25 .The rate of interest is :
a) 3% b) 4% c) 5% d) 6%
3)The least number of complete years in which a sum of money put out at 20% compound interest will be more than doubled is :
a) 5 years b) 4 years c) 6 years d) 7 years
4)At compound interest , if certain sum of money doubles in n years , then the amount will be four fold in :
a) 2n^2 years b) n^2 years c) 4n years d) 2n years
5)A saving bank gives interest which compounds annually. Mr. X deposited Rs. 100 and received Rs. 121 at the end of second years . Rate of compound interest per annum is :
a) 10% b) 20% c) 10.5% d) 20.5%
6) The difference between simple and compound interest compounded yearly at the rate of 4% in a two years period on a certain amount of money is Rs. 1. The amount of money is :
a) Rs. 450 b) Rs. 575 c) Rs. 600 d) Rs. 625
7) A sum of money on compound interest amounts to Rs. 9680 in 2 years and to Rs. 10648 in 3 years . the rate of interest per annum is :
a) 5% b) 10% c) 15% d) 20%
8)The difference between compound interest and simple interest at the same rate for Rs. 5000 for 2 years is Rs. 72 . The rate of interest per annum is :
a) 6% b) 8% c) 10% d) 12%
9) A sum of money amounts to Rs. 6690 after 3 years and to Rs. 10035 after 6 years on compound interest . The sum is :
a) Rs. 4400 b)Rs. 4460 c) Rs. 4520 d) Rs. 4445
10) A sum amounts to Rs. 2916 in 2 years and to Rs. 3149.28 in 3 years at compound interest. The sum is :
a) Rs. 1500 b)Rs.2000 c) Rs. 2500 d) Rs. 3000
Answer
1) d 2) c 3) b 4) d 5) a 6) d 7) b 8) d 9) b 10) c
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