Compound Interest
When we deposit
some money in a bank, every year some interest is added to it. The interest is
not the same for every year and it goes on increasing year by year. This is
because at the end of the first year, simple interest is calculated and added
to the principal to get the amount. This amount becomes the principal for the
next year.
At the end of the
second year again, the amount is calculated by adding principal and interest.
This amount becomes the principal for the third year and so on. Every year, the
principal changes and the interest is calculated on the amount of the previous
year. When interest is calculated in this manner, we call it compound
interest.
Derivation of Compound Interest Formula
When the time period is longer,
calculating simple interest every time and adding it to the principal amount to
get the principal for next year is very complex calculation and time consuming.
Using step by
step procedure, we will arrive at a formula to find the amount and the compound
interest.
Let us take the
principal as ₹ P, rate = R% p.a. and time = T years
What is P, R and T in Compound Interest Formula?
We know that the formula to calculate compound interest is as
follows:
In this formula, P is the initial principal for which the compound
interest is to be calculated.
R is the rate of interest. It is given in percentage. Suppose the
rate is 5% per annum. It means that on a deposit of ₹ 100 or on a loan of ₹
100, you will receive ₹ 5 or pay ₹ 5 as an interest in one year.
What is Conversion Period?
Sometimes, interest is calculated and added to the principal after
every 6 months or 3 months. This is called interest is compounded half-yearly
or quarterly.
The time period T after which the interest is added to the
principal each time to form a new principal is called the conversion period.
This period may be 1 year, 6 months or
3 months. According to that, calculations are done by reducing the rate of interest
half-yearly or quarterly.
Time Period and Rate When the Interest is Compounded Half-yearly and Quarterly
If the interest
is compounded half-yearly, there are two conversion periods in a year and the
rate of interest is half the annual rate.
Let us suppose, a
sum of ₹ 10,000 is borrowed for 1 year 6 months at 8% p.a. compounded
half-yearly.
In this case, P =
₹ 10,000
In 1 year and 6
months, there are 3 half years, hence we have 3 conversion periods; or T =
3
Rate (R) = half
of the annual rate = ½ × 8% = 4%
Therefore, A = P(1
+ R/100)T
= 10,000(1 + 4/100)3
= 10,000 (1 + 0.04)3
= 10,000 (1.04)3
= 10,000 × 1.04 × 1.04
× 1.04
= ₹ 11,248.64
Thus, CI = ₹ 11,248.64
– ₹ 10,000
= ₹ 1,248.64
If the interest
is compounded quarterly, there are 4 conversion periods in a year. The interest
rate is one-fourth of the annual rate.
For example, if a sum of ₹ 10,000 is borrowed for 1 year at 8% p.a.
compounded quarterly, the number of conversion periods is 4 quarters and the
rate = ¼ × 8% = 2% quarterly
Therefore, A = P (1
+ R/100)T
= 10,000 (1 + 2/100)4
= 10,000 (1 + 0.02)4
= 10,000 (1.02)4
= 10,000 × 1.02 × 1.02
× 1.02 × 1.02
= ₹ 10,824.32
Thus, CI = ₹ 10,824.32
– ₹ 10,000
= ₹ 824.32
Example 1: Find the amount and compound interest on ₹ 2400 for 2 years
at 10% p.a. compounded half-yearly.
Solution:
Here, P = ₹ 2400, R = 10% p.a. = 10/2 = 5%
half-yearly, T = 2 years = 4 half-years
Amount (A)
= P (1 + R/100)T
= 2400 (1 + 5/100)4
= 2400
(1 + 0.05)4
= 2400 (1.05)4
= ₹ 2917.21
Compound interest = A – P = ₹ 2917.21 – ₹ 2400 = ₹ 517.21
Hence, amount = ₹ 2917.21 and compound interest = ₹ 517.21
Example 2: Find the amount and compound interest on ₹ 12,000 for 9
months at 16% p.a., interest being compounded quarterly.
Solution:
Here, P = ₹ 12,000, R = 16% p.a. = 16/4 = 4% quarterly,
T = 9 months = 3 quarters
Amount (A)
= P (1 + R/100)T
= 12,000 (1 + 4/100)3
= 12,000 (1 + 0.04)3
= 12,000 (1.04)3
= ₹ 13,498.37
Compound interest = A – P = ₹ 13,498.37 – ₹ 12,000 = ₹ 1498.37
Hence, amount = ₹ 13,498.37 and compound interest = ₹ 1498.37
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