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Compound InterestWhen the borrower X and the lender Y agrees to fix up a certain
time for example yearly, half yearly or quarterly to settle the
previous money, then the difference between the amount and the
money borrowed is said to be the Compound Interest and it
denoted by C.I. In these calculations, principal for the second unit
of time is the amount of first unit of time and so on.
In compound interest, the interest for each period is added to the
principle before interest is calculated for the next period. With
this method the principle grows as the interest is added to it.
This method is mostly used in investments such as savings
account and bonds.
Basic Formulas of Compound Interest
If A = Amount
P = Principle
C.I. = Compound Interest
T = Time in years
R = Interest Rate Per Year
Amount Due at the end of the time period is given byf
Where:
P: Principal (original amount)
R: Rate of Interest (in %)
T: Time period (yearly, half-yearly etc.)
The Compound Interest over the time period T is given by by
the formula:
This can be written as:
Compound Interest= P {(1+r/100)t
-1}
To understand compound interest clearly, let’s take an example.
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1000 is borrowed for three years at 10% compound interest.
What is the total amount after three years?
Year Principle Interest (10%) Amount
1st 1000 100 1100
2nd 1100 110 1210
3rd 1210 121 1331
After three years,
In simple interest, the total amount would be 1300
And in compound interest, the total amount would be 1331.
Shortcut formulas for compound interestRule 1: If rate of interest is R1% for first year, R2% for
second year and R3% for third year, then
Rule 2:
If principle = P, Rate = R% and Time = T years then
1. If the interest is compounded annually:
2. If the interest is compounded half yearly (two times in year):
3. If the interest is compounded quarterly (four times in year):
Rule 3:
If difference between Simple Interest and Compound
Interest is given.
If the difference between Simple Interest and Compound
Interest on a certain sum of money for 2 years at R% rate is
given then
If the difference between Simple Interest and Compound
Interest on a certain sum of money for 3 years at R% is given
then
Rule 4:
If sum A becomes B in T1 years at compound interest, then
after T2 years
Look up Table
4% 5% 10% 20%
1 year
1
100
r
26/25 21/20 11/10 6/5
2 year
2
1
100
r
676/625 441/400 121/100 36/25
3 year
3
1
100
r
17576/15625 9261/8000 1331/1000 216/125
Installment paid with compound interestTo calculate the installments paid with compound interest, we
use the following formula
Where,
P= Principal
R=rate
n= number of installments
×= Amount of installment
Example:
A sum of Rs. 1275 is borrowed at 4% pa compound interest and
paid back in 2 equal annual installments. What is the amount of
each installment?
Solution:
Let the value of installment be ×
Equating the amounts
1275×(1.04)2=× +1.04×
×= Rs. 676
Example:
A sum of Rs. 550 is to be repaid in 2 equal annual installments.
If rate =20% compounded annually, then the value of each
installment will be?
Solution:
Let the value of installment be ×
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Equating the amounts
550*(1.2)2
=×+1.20×
×= Rs. 360
Example
A Sum of Rs. 2600, is lent out in two parts S.I. at 10% for 5
yr is equal to S.I. on 2nd part at 9% rate for 6 yr. find the
ratio of parts.
Solution :
Given SI1 = SI 2
P1 : P2 = 1/R1T1 : 1/R2T2
= 1/(10*5) : 1/(9:6)
27:25
Example 1:
Maninder took a loan of Rs. 10000 from Prashant. If the rate of
interest is 5% per annum compounded annually, find the amount
received by Prashant by the end of three years
Solution:
The following is the data given:
Principal, P= 10000
Rate = 5%
Time =3 years
Using the formula for Compound Interest:
A = P(1+R/100)t
So A= 10000(1+5/100)3
A = 10000(1+1/20)3
A = 10000 x 21/20 x 21/20 x21/20 =11576.25
So, the total amount paid by Maninder at the end of third year is
Rs.11576.25
Example 2:
Richa gave Rs. 8100 to Bharat at a rate of 9% for 2 years
compounded annually. Find the amount of money which she
gained as a compound interest from Bharat at the end of second
year.
Solution:
Principal value = 8100
Rate = 9%
Time = 2 years
So the total amount paid by Bharat
= 8100(1+9/100)2
=Rs. 9623.61
The question does not probe the amount, rather, it wants to
know the CI paid, that the difference between the total amount
and original principal.
The Compound Interest = 9623.61 –8100 = 1523.61
Example 3:
The difference between compound interest and simple
interest is 2500 for two years at 2% rate, then find the
original sum.
Solution:
Given difference is = 2500
So, Simple Interest = (P X R X T)/100
Compound Interest = P [(1+R/100)t
–1]
So the difference between both of them is
= P[(1+R/100)T
-1] – PRT/100
= P [(1+R/100)T
-1-RT/100]
2500 = P [{(1+2/100)2
-1}-4/100]
On simplification this equation, the sum will be = Rs. 6250000
Let’s use the shortcut method:
When timegiven is 2 years, Difference = P(R/100)2
Since the difference given is Rs 2500, we have
2500 = P (2/100)2
=> 2500 = P (1/50)2
=> 2500= P (1/2500)
=> 6250000=P
So, the sum is Rs.6250000.
Question 1:
An amount of Rs 1000 is borrowed at CI at the rate of 2% per
annum. What will be the amount to be paid after 3 years if the
interest is compounded annually?
A. 926.24
B. 1248.34
C. 1061.28
D. 1678.34
Answer and Explanation
Option C.
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Question 3:
The difference between the compound interest and the simple
interest on a certain sum at 4% rate for 2 years is Rs 100. What
will be the amount invested?
A. 45500
B. 52500
C. 62000
D. 62500
Option D
The difference between compound interest and simple interest
for two years is given by
Question 4:
The difference between the compound interest and the simple
interest on a certain sum at 4% rate for 3 years is Rs 100. What
will be the amount invested?
A. 20559
B. 25559
C. 16559
D. 28559
Option A
The difference between CI and SI for three years is given by
Question 5:
A sum of money invested at compound interest amounts to Rs.
650 at the end of first year and Rs. 676 at the end of second
year. The sum of money is:
A. Rs. 600
B. Rs. 540
C. Rs. 625
D. Rs. 560
Option C
Simple Interest for one year = Compound Interest for one
year
Interest on Rs. 650 for 1 year = 676 –650 = Rs. 26
A sum at a rate of interest compounded yearly becomes Rs.
A1
in n years and Rs. A2
in (n + 1) years, then
Example-3:
A sum of money invested at compound interest amounts to Rs.
100 at the end of first year and Rs. 120 at the end of second
year. The sum of money is :
Solution:
Simple Interest for one year = compound interest for one
year
Interest on Rs. 100 for 1 year = 120-100= Rs. 20
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If an amount of money grows up to Rs x in t years and up to
Rs y in (t+1) years on compound interest, then
Derivation for this result:
Principal + CI for t years = x …… (1)
Principal + CI for (t+1) years= y ……. (2)
(2) –(1) =>CI for last year = y-x
Which is basically the simple interest upon x
Example-2:
An amount of money grows upto Rs 3000 in 3 years and upto
Rs 4000 in 4 years on compound interest. What will be the rate
percent?
Solution:
Principal + CI for 3 years = 3000 …… (1)
Principal + CI for 4 years= 4000 ……. (2)
Hence (2) –(1) =>CI for 4th year = 4000-3000= Rs 1000
Which is basically the simple interest upon 3000
If a certain sum becomes x times of itself in t years, the rate
of compound interest will be equal to
Example 4:
If a certain sum becomes 16 times in 2 years , what will be the
rate of compound interest?
Solution:
Using the formula derived above:
If the compound interest on a certain sum for 2 years is CI
and simple interest for two years is SI ,then rate of interest
per annum is
If the compound interest on a certain sum for 2 years is CI
and simple interest for two years is SI ,then rate of interest
per annum is
Example 5:
If the compound interest on a certain sum for 2 years is 20rs and
simple interest for two years is 10rs ,then what wil be the rate
of interest per annum ?
Solution:
Using the formula derived above:
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SOLVED EXERCISE
1. A bank offers 5% compound interest calculated on half-yearly
basis. A customer deposits Rs. 1600 each on 1st January and 1st
July of a year.
At the end of the year, the amount he would have gained by way
of interest is:
(A) 123 (B) 122
(C) 121 (D)120
2. The compound interest on Rs. 30,000 at 7% per annum is Rs.
4347. The period (in years) is:
(A) 2.5 B) 2
(C) 3 D) 4
3. At what rate of compound interest per annum will a sum of Rs.
1200 become Rs. 1348.32 in 2 years?
(A) 8 % (B) 9%
(C) 6 % (D) 8.5 %
4. The difference between simple interest and compound on Rs.
1200 for one year at 10% per annum reckoned half-yearly is:
(A) Rs. 3 (B) Rs. 4
(C) Rs. 3.5 (D) Rs. 7.5
5. 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) 4 (B) 5
(C) 6 (D) 2.5
6. What will be the compound interest on a sum of Rs. 25,000 after
3 years at the rate of 12 p.c.p.a.?
(A) Rs.10123.20 (B) Rs. 9000
(C) Rs. 12000 (D) Rs. 10163.34
7. Simple interest on a certain sum of money for 3 years at 8% per
annum is half the compound interest on Rs. 4000 for 2 years at
10% per annum. The sum placed on simple interest is:
(A) Rs. 1650 (B) Rs. 2000
(C) Rs. 1750 (D) Rs.1550
SOLVED EXERCISE EXPLANATION
1. Amount = Rs. [1600×(1+ 5/200)^2 + 1600 × (1+5/200)]
= Rs. 3321
So CI = Amount- Principal
= Rs. 3321 – Rs. 3200 = Rs. 121
2. Amount = Rs. (30000 + 4347) = Rs. 34347,
Let the time be n years then
30000(1+7/100) ^n = 34347
(107/100) ^n = 34347/30000
So n= 2 year.
3. Let rate r % per annum
1200× (1+r/100) ^ 2 = 1348.32
(1+r/100) ^ 2 = 1348.32/1200
1+r/100 = 106 / 100, r= 6 %
4. SI =Rs. (1200 ×10×1)/100= Rs. 120
CI = Rs.[ 1200×(1+5/100) ^2 -1200] = Rs.123
So CI-SI = Rs. 3
5. P(1+20/100) ^n > 2P
(6/5)^ n >2
(6/5×6/5×6/5×6/5)>2
so n = 4 years
6. Amount= Rs. 25000(1+12/100)^3= 35123.20
So CI= Rs. (35123.20 -25000) = Rs. 10123.20
7. C.I.= Rs. 4000(1+10/100)^2 –40
= Rs. 840
Sum= Rs. (420 × 100)/(3×8) = Rs. 1750
UNSOLVED EXERCISE
(1) The compound interest on a certain sum for 2 years is Rs. 786
and S.I. is Rs. 750. If the sum is invested such that the S.I. is Rs.
1296 and the number of years is equal to the rate per cent per
annum, Find the rate of interest?
(2) Hari took an educational loan from a nationalized bank for his 2
years course of MBA. He took the loan of Rs.5 lakh such that he
would be charged at 7% p.a. at CI during his course and at 9%
CI after the completion of the course. He returned half of the
amount which he had to be paid on the completion of his studies
and remaining after 2 years. What is the total amount returned
by Hari?
(3) Suresh borrows Rs.6375 to be paid back with compound interest
at the rate of 4 % pa by the end of 2 year in two equal yearly
installments. How much will each installment will be?
(4) A part of 70000 is lent out at 10% annum. The rest of the
amount is lent out at 5% per annum after one year. The ratio of
interest after 3 years from the time when first amount was lent
out is 1:2. Find the second part that was lent out at 5%.
(5) During the first year the population of a village is increased by
5% and the second year it is diminished by 5%. At the end of
the second year its population was 31500. What was the
population at the beginning of the first year?
(6) Karthik lends a certain amount to Vignesh on simple interest for
two years at 20%. Vignesh gives this entire amount to Kamal on
compound interest for two years at the same rate annually. Find
the percentage earning of Vignesh at the end of two years on the
entire amount.
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(7) The difference between the total simple interest and the total
compound interest compounded annually at the same rate of
interest on a sum of money at the end of two years is Rs. 450.
What is definitely the rate of interest per cent per annum?
(8) Rahul saves an amount of 800 every year and then lent that
amount at an interest of 10 percent compounded annually. Find
the amount after 3 years.
(9) A man borrows 2000 rupees at 10% compound interest. At the
end every year he pays rupees 1000 back. How much amount
should he pay at the end of the third Year to clear all his debt?
(10) The difference between interest received by Vivek and Vimal is
Rs.405 on Rs.4500 for 3 years. What is the difference in rate of
interest ?
(11) A sum of rupees 3200 is compounded annually at the rate of 10
paisa per rupee per annum. Find the compound interest payable
after 2 years.
(12) What sum of money will amount to rupees 1124.76 in 3 years, if
the rate of interest is 5% for the first year, 4% for the second
year and 3% for the third year?
(13) A sum of rupees 3903 is divided between P and Q such that the
share of P at the end of 8 years is equal to the share of Q after 10
years. Find the share of P if rate of interest is 4% compounded
annually.
(14) A sum of money is lent for 2 years at 10% p.a. compound
interest. It yields Rs 8.81 more when compounded semiannually than compounded annually. What is the sum lent?
(15) On a certain sum of money, after 2 years the simple interest and
compound interest obtained are Rs 400 and Rs 600 respectively.
What is the sum of money invested?
(16) The difference between CI and SI on a sum for 2 years at 10%
per annum when the interest is compound annually is Rs 16. If
the interest were compounded half yearly the difference in the
interest will be
(17) The simple interest accrued on an amount of Rs. 14,800 at the
end of three years is Rs. 6,216. What would be the compound
interest accrued on the same amount at the same rate in the same
period?
(18) Uday invested Rs 20,000 with rate of interest 20% per annum.
The interest was compounded half yearly for first one year and
in the next year it was compounded yearly. What will be the
total interest earned at the end of two years?
(19) Find the compound interest on Rs. 64,000 for 1 year at the rate
of 10% per annum compounded quarterly (to the nearest
integer).
(20) The difference between C.I. and S.I. on a certain sum of money
at 10% per annum for 3 years is Rs. 620. Find the principal if it
is known that the interest is compounded annually.
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