8. A particular amount deposited for a
particular rate of interest compounded
annually amount to Rs. 375 in 4 year and to
Rs. 390 in 5 years. What is the rate of
interest?
(A) 1%
(B) 4%
(C) 5%
(D) 6%
(E) 7.5%
Answers
Answer:
Compound Interest
You have learned about the simple interest and formula for calculating simple interest and amount. Now, we shall discuss the concept of compound interest and the method of calculating the compound interest and the amount at the end of a certain specified period. We shall also study the population growth and depreciation of the value of movable and immovable assets.
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Compound Interest
If the borrower and the lender agree to fix up an interval of time (say, a year or a half year or a quarter of a year etc) so that the amount (Principal + interest) at the end of an interval becomes the principal for the next interval, then the total interest over all the intervals, calculated in this way is called the compound interest and is abbreviated as C.I.
Compound interest at the end of a certain specified period is equal to the difference between the amount at the end of the period and original principal i.e. C.I. = Amount – Principal. In this section, we shall discuss some examples to explain the meaning and the computation of compound interest. Compound interest when interest is compounded annually.
Compound Interest
Example 1
Find the compound interest on Rs 1000 for two years at 4% per annum.
Solution: Principal for the first year =Rs 1000
S
I
=
P
×
R
×
T
100
S
I
f
o
r
1
s
t
y
e
a
r
=
1000
×
4
×
1
100
S
I
f
o
r
1
s
t
y
e
a
r
=
R
s
40
Amount at the end of first year =Rs1000 + Rs 40 = Rs 1040. Principal for the second year = Rs1040
S
I
f
o
r
2
n
d
y
e
a
r
=
1040
×
4
×
1
100
S
I
f
o
r
2
n
d
y
e
a
r
=
R
s
41.60
Amount at the end of second year,
A
m
o
u
n
t
=
R
s
1040
+
R
s
41.60
=
R
s
1081.60
Therefore,
C
o
m
p
o
u
n
d
i
n
t
e
r
e
s
t
=
R
s
(
1081.60
–
1000
)
=
R
s
81.60
Remark: The compound interest can also be computed by adding the interest for each year.
Browse more Topics under Compairing Quantities
Comparison Using Percentage
Uses of Percentage
Discount and Commissions
Growth and Depreciation
Profit and Loss
Ratio and Percentage
Tax
Compound Interest when Compounded Half Yearly
Example 2: Find the compound interest on Rs 8000 for 3/2 years at 10% per annum, interest is payable half-yearly.
Solution: Rate of interest = 10% per annum = 5% per half –year. Time = 3/2 years = 3 half-years
Original principal = Rs 8000.
I
n
t
e
r
e
s
t
f
o
r
t
h
e
f
i
r
s
t
h
a
l
f
y
e
a
r
=
8000
×
5
×
1
100
=
420
. Amount at the end of the first half-year= Rs 8000 +Rs 400 =Rs8400
Principal for the second half-year =Rs 8400
I
n
t
e
r
e
s
t
f
o
r
t
h
e
s
e
c
o
n
d
h
a
l
f
−
y
e
a
r
=
8400
×
5
×
1
100
=
420
Amount at the end of the second half year = Rs 8400 +Rs 420 = Rs 8820
I
n
t
e
r
e
s
t
f
o
r
t
h
e
t
h
i
r
d
h
a
l
f
y
e
a
r
=
8820
×
5
×
1
100
=
R
s
441
Amount at the end of third half year= Rs 8820+ Rs 441= Rs 9261. Therefore, compound interest= Rs 9261- Rs 8000= Rs 1261. Therefore,
c
o
m
p
o
u
n
d
i
n
t
e
r
e
s
t
=
R
s
9261
−
R
s
8000
=
R
s
1261
Compound Interest by Using Formula
In this section, we shall obtain some formulae for the compound interest.
Case 1
Let P be the principal and the rate of interest be R% per annum. If the interest is compounded annually, then the amount A and the compound interest C.I. at the end of n years is given by:
A
=
P
(
1
+
R
100
)
n
and
C
I
=
A
−
P
C
I
=
P
(
1
+
R
100
)
n
−
P
C
I
=
P
[
(
1
+
R
100
)
n
−
1
]
Example 3: Find the compound interest on Rs 12000 for 3 years at 10% per annum compounded annually.
Solution: P =Rs 12000, R =10% per annum and n=3. Therefore, amount (A) after 3 years
A
=
P
(
1
+
R
100
)
3
A
=
12000
(
1
+
10
100
)
3
A
=
12000
(
11
10
)
3
A
=
12000
(
11
10
)
×
(
11
10
)
×
(
11
10
)
A
=
15972
C
o
m
p
o
u
n
d
i
n
t
e
r
e
s
t
=
A
−
P
C
I
=
R
s
15972
−
R
s
12000
=
R
s
3972
Case 2
When the interest is compounded half-yearly.
A
=
P
(
1
+
R
200
)
2
n
C
I
=
P
[
(
1
+
R
200
)
2
n
−
1
]
Case 3
When the interest is compounded quarterly.
A
=
P
(
1
+
R
400
)
4
n
C
I
=
P
[
(
1
+
R
400
)
4
n
−
1
]
Case 4
Let be the principal and the rate of interest be R1% for the first year, R2% for second year, R3% for third year and so on and last Rn% for the nth year . Then the amount (A) and the compound interest C.I. at the end of n years are given by:
A
=
P
(
1
+
R
1
100
)
(
1
+
R
2
100
)
(
1
+
R
3
100
)
A = P (1+R1/100)(1+R2/100)…(1+Rn/100) and
C
I
=
A
−
P
respectively.
Case 5
Let p is the principal and the rate of interest is R% per annum. If the interest is compounded annually but time is the fraction of a year, say 21/4 years, then amount A is given by:
A
=
P
(
1
+
R
100
)
5
(
1
+
R
4
100
)
and
C
I
=
A
−
P