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Compound interest is a type of interest which is applied on the Principal and Interest over a given period of time. The interest gathered on a principal over a time period is also considered under the Principal. It is a new method of calculating interest which is majorly used in financial and business transactions. The power of compounding can be understood, when we try to observe the compound interest values gathered over successive time periods.
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Compound Interest
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Compound interest is paid on both principal and interest which is compounded at regular intervals. At regular intervals, the interest so far gathered and clubbed with the existing principal amount and then for the new principal, interest is calculated. The new principal should be the sum of the initial principal and the interest accumulated so far.
Compound Interest = Interest on Principal + Compounded interest at regular intervals
Compound Interest
Compound interest is calculated annually, semi-annually, quarterly, monthly etc. Banks or financial organizations calculate the amount for loans on the basis of compound interest only.
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Compound interest Formula Detailed Video Explanation:
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Compound Interest Formula
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Compound interest is calculated after finding the total amount over a period of time, on the basis of rate of interest and initial principal. For initial principal of “P”, rate of interest per annum “r”, time period “t” in years, frequency of number of times interest is compounded “n”, formula for calculation of compound amount “A” is -
A = P(1+r/n)nt
The above formula represents the total amount at the end of the time period and includes compounded interest and principal. Therefore, we can find the compound interest by subtracting principal from this amount.
C.I = P(1+r/n)nt - P
where, C.I = Compound Interest
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Terms Related to Compound Interest
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- Principal - Sum of money lent for a certain period for a certain rate of interest.
- Time - Duration for which the principal is mainly calculated in years.
- Interest - Profit earned imparting a principal for a certain period.
- Rate - It is the percentage of interest and lending a sum of money.
- Amount - The final money left after all calculations is the amount. It is the sum of original principal and total compound interest earned.
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Derivation of Compound Interest Formula
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Before starting derivation, let us understand the difference between simple and compound interest. For simple interest, the principal remains constant throughout the time period but for compound interest, the interest is added to the principal after every regular interval.
Derivation
Let us consider the principle as “P” and rate of interest as “R”.
At the end of first compounding period, the simple interest on principal is P*r/100.
Hence, the amount is P+P*r/100 = P(1+r/100).
The amount will be considered as principal for the second computation period.
After the end of second compounding period, the simple interest on principal is:
P(1+r/100) x (r/100)
Hence the amount is P(1+r/100)x(r/100) + P(1+r/100) + P(1+r/100)x(r/100) = P(1+r/100)2.
Following the above formula for “n” compounding periods, the amount at the end of the nth compounding period is A = P(1+r/100)n
From the above computations, we can observe clearly that compound interest will be the same as simple interest for the first interval. But over a period of time, there exists a difference in returns.
The simple interest for each of the years will be the same, as the principal calculated will be constant. But the compound interest varies each year. The principal for a particular year is equal to the sum of initial principal value, and accumulated interest of the past years.
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Compound Interest Formula for different Time periods
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Compound Interest formulas for different time periods are given below.
Half Yearly
Formula to calculate compound interest when principal is compounded semi-annually or half-yearly is given as -
C.I = P(1+r/2/100)2T - P
Formula to calculate amount when principal is compounded semi-annually or half-yearly is given as -
A = P(1+r/2/100)2T
Quarterly
Formula to calculate compound interest when principal is compounded quarterly is given as -
C.I = P(1+r/4/100)4T - P
Formula to calculate amount when principal is compounded semi-annually or half-yearly is given as -
A = P(1+r/4/100)4T
Monthly
Formula to calculate compound interest when principal is compounded monthly is given as -
C.I = P(1+r/12)12T - P
Daily
Formula to calculate compound interest when principal is compounded daily is given as -
C.I = P(1+r/365)365T - P
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Things to Remember
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- While finding the compound interest, rate of interest and each time period must be of the same duration.
- Banks, insurance companies and other financial organizations levies compound interest.
- Compound interest depends upon the amount gathered at the end of the previous tenure but not on the original principal.
- The units of compound interest are units of currency and same as the units used for the principal values. If the principal is in rupees, dollars then, compound interest will also be in rupees, dollars respectively.
- We should need to know the principal, time period, rate of interest, and time interval to calculate the compound interest.
- The value of compound interest can be greater than the principal. It’s value increases and varies for successive periods.
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Sample Questions
Ques. Calculate the compound interest on Rs 10000 in 2 years at 4% per annum, the interest being compounded yearly. (2 Marks)
Ans. Rate of interest = 4%
Applying net% effect formula for effective rate of compound interest for 2 years, we get
Net% effect = x+y+xy/100%
x = y = 4%
= 4+4+4*4/100 = 8+0.16 = 8.16%
C.I = 8.16% of 10,000
C.I = 8.16*10000/100 = Rs 816
Ques. Ramesh lends Rs 4000 to Suresh at an interest rate of 10% per annum, compounded half-yearly for a period of 2 years. How much does he get after a period of 2 years from Suresh? (2 Marks)
Ans. Principal(P) = Rs 4000
Rate of interest(r) = 10%
Conversion period is half year so r will be 10/2 = 5%
Time period(t) = 2 years
Substituting the values in compound interest formula,
Amount(A) = P(1+(r/2)/100)2n
A = 4000(1+(10/2)/100)2(2)
A = Rs 4862.03
Therefore, the final amount is Rs 4862.03 which he gets after a period of 2 years.
Ques. Calculate the compound interest if Rs 5000 as Principal amount is invested for 2 years at 10% p.a. compounded half-yearly. (3 Marks)
Ans. We know that, A = P(1+r/100)n
P = 5000
r = 10%
n = 2 years
Substituting values for calculating compound interest,
A = 5000(1+10/100)2
A = 5000(1+0.1)2
A = 5000(1.1)2
A = 5000*1.21
A = Rs 6050
C.I = A - P
C.I = 6050 - 5000
C.I = Rs 1050
So, Rs 1050 is levied if Rs 5000 was invested for 2 years at 10% p.a. compounded half-yearly.
Ques. The Compound interest on a sum of Rs 1000 in 2 years is Rs 440. Find rate of interest. (3 Marks)
Ans. P = Rs 1000
n = 2 years
C.I = Rs 150
Substituting the values in the formula, C.I = A - P
440 = A - 1000
A = 1000 + 440
A = 1440
So, Amount comes out to be Rs 1440.
Now, we calculate rate of interest(r) by using formula, A = P(1+r/100)n
Substituting values, 1440 = 1000(1+r/100)2
1440/1000 = (1+r/100)2 (Taking Square root both sides)
12/10 = 1+r/100
12/10 - 1 = r/100
(12-10)/10 = r/100
2/10 = r/100
r = 20%
So, Rate of interest is 20% for a sum of Rs 1000 in 2 years.
Ques. The difference between Simple Interest and Compound Interest for a period of 2 years at 10% per annum is Rs 15. Calculate the principal? (2 Marks)
Ans. We know that formula difference = P(r/100)2
15 = P(10/100)2
15 = P(100/1000)
15 = P(1/100)
Therefore, Principal = Rs 1500.
Ques. A man deposited Rs 100000 in a bank. In return, he got Rs 133100. Bank gave interest 10% per annum. How long did he keep the money in the bank? (3 Marks)
Ans. Principal = Rs 100000
A = Rs 133100
r = 10%
n = ?
Formula to calculate time period, A = P(1+r/100)n
133100 = 100000(1+10/100)n
133100/100000 = (1+10/100)n
(11/10)3 = (110/100)n
(11/10)3 = (11/10)n
So, we will compare powers, n = 3.
Therefore, the man has to keep his money for 3 years in the bank.
Ques. A certain sum amounts to Rs 7200 for a period of 2 years at 6% per annum compound interest, which is compounded annually. Calculate the principal for the problem. (3 Marks)
Ans. A = Rs 7200
n = 2 years
r = 6%
By using the formula, A = P(1+r/100)n
7200 = P(1+6/100)2
7200 = P(106/100)2
7200 = P(1.1236)
P = 7200/1.1236
P = Rs 6407
Therefore, the principal is Rs 6407 for the amount Rs 7200.
Ques. For 50% increase in an amount in 5 years at simple interest, then Calculate the compound interest of Rs 12000 after 3 years at the same rate. (2 Marks)
Ans. In Simple interest,
Let P = 100, I = 50, T = 5 years
R = 50*100/100*5 = 10%
In Compound interest,
P = 12000, T = 3 years, R = 10%
C.I = 12000*(1+10/100)3 - 1
C.I = Rs 3972
Therefore, the compound interest will be Rs 3972 for Rs 12000 after 3 years at the same rate.
Ques. Rahul and Vaishnavi invested Rs 10000 each in scheme A and scheme B respectively for 3 years. Scheme A offers simple interest at 12% per annum and scheme B offers compound interest at 10% per annum. After 3 years, who will have a larger amount and by how much? (3 Marks)
Ans. First, calculate the total rate % that Rahul will have after 3 years.
Rahul invested at 12% per annum simple interest
So for a tenure of 3 years, he will get = 12*3 = 36%
Vaishnavi invested at 10% per annum compound interest
By net% effect formula, we can calculate total % for 3 years period =
Net% effect = x+y = xy/100%
For first 2 years, here x = y = 10%
= 10 + 10 = 10*10 = 21/100%
And for next year, here x = 21% and y = 10%
= 21 + 10 = 21*10 = 33.1%
So, the difference between S.I and C.I comes out to be 36% - 33.1% = 2.9% (S.I is more)
Hence, Rahul will get 2.9% of 10000 = Rs 290
So, Rahul will have Rs 290 more than Vaishnavi.
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