Contents:
Compound Interest
Click here to learn more about Simple Interest.
Compound Interest:
Calculation of Compound Interest is a little different than the Simple Interest.
Calculating the Interest on the Amount to be received, in certain interval of time is called Compound Interest.
This means, compound interest is calculated on the sum of Principal and Interest, of previous time, which is our new Principal.
Let's say, I have deposited some sum 'x' in a bank and I received 'y' interest after one year. I don't want to take the Interest out rather I want bank to pay me interest on my new Principal 'z' that is 'x+y'.
Here all three variables, Principal, Rate of Interest and Duration of time can vary in each condition.
Things to remember:
- While calculating the Interest for the First time, value of Principal is equal to the initial Principal.
- In Compound Interest, the value of the Principal of current time is equal to the value of Amount of previous time.
- The duration of Time can be daily, monthly, quarterly, half-yearly, yearly or other time intervals.
- In Compound Interest, the Rate of Interest might also vary in different time interval.
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Now, let us create the formulae (DERIVATION):
Note: R% = R/100
Let,
Initial Principal = Rs P
Rate of Interest = R%
Duration of Time = T years
Now,
Interest for the first year
= P * T * R%
= P*1*R%
= PR%
So, the amount in the first year
= Principal + Interest
= P + PR%
= P (1 +R%) [Taking 'P' common]
We have,
Principal for second year
= Amount in first year
= P(1 +R%)
And, Interest for the second year
= P * T * R%
= P(1 +R%) * 1 * R%
= P(1 +R%) R%
So, the amount in the second year = Principal of 2nd year + Interest of second year
= P(1 +R%) + P(1 +R%)R%
= (1+ R%) (P+ PR%)
= (1+ R%) P(1 +R%)
= P (1 +R%)(1 +R%)
= P (1 +R%)²
For third year,
Principal for third year= Amount of second year = P (1 +R%)²
And, Interest for the third year = P (1 +R%)² * 1* R%
= P (1 +R%)² R%
So, Amount in the third year = Principal of third year + Interest of Third Year
= P (1 +R%)² + P (1 +R%)²R%
= P (1+ R%)² + (1 + R%)
= P (1 +R%)³
From above results,
We can conclude that:
Compound Amount in 'T' Years = P (1 +R/100)^T
Also,
Amount = Principal + Interest
So, Interest = Amount - Principal
Interest in 'T' Years = P (1 +R%)^T - P
Compounded Interest = = P [(1 +R/100)^T -1]
Calculation of Compounded Interest but the rate of interest being different in every year:
Let us have;
Initial Principal = Rs P
Rate of Interest for first year = R1%
Rate of Interest for second year = R2%
Rate of Interest for third year = R3%
Now,
Interest for First Year
= P * 1 * R1%
= P R1%
Amount in First Year
= P + P R1%
= P (1 +R1%)
And,
Interest for Second Year
= P (1 +R1%) * 1 * R2%
= P (1 +R1%) R2%
Amount in Second Year
= P (1 +R1%) + P (1 +R1%) R2%
= P (1 +R1%) (1 +R2%)
Also,
Interest for Third Year
= P (1 +R1%) (1 +R2%) * 1 * R3%
= P (1 +R1%) (1 +R2%) R3%
Amount in Third Year
= P (1 +R1%) (1 +R2%) + P (1 +R1%) (1 +R2%) R3%
= P (1 +R1%) (1 +R2%) (1 +R3%)
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Therefore,
Compound Amount= P (1 +R1/100) (1 +R2/100) (1 +R3/100)Compound Interest= P (1 +R1/100) (1 +R2/100) (1 +R3/100) - P= P [(1 +R1/100) (1 +R2/100) (1 +R3/100) -1]
Calculation of Compounded Interest when the Rate of Interest is the same but the duration of time is given in T years and M months:
First of all,
There are 12 months in a year and 365 days in a year.
We have the formula:
Interest
= P*T*R%
What if we multiply and divide T by 12?
= P*T*12/12*R%
= P*T*R%
There is no change in the equation. This is because there are twelve months in one year. So, if we divide 12 months by 12 months, we get 1 year.
Similarly, in this formula:
Amount = P (1 +R/100)
There is a secret T in the formula which is equal to 1.
Let's write this:
Amount = P (1 + 1*R/100)
Converting the " T =1 year" into months, we get; 1 = 12
To write this, we need to multiply and divide 1 by 12 so that it doesn't affect the equation;
Amount = P (1 + 1*12/12* R/100)
or, Amount = P (1 + 12R/1200)
Now, in above equation, put 12 as 'M' (variable) that will represent the total months:
Amount = P (1 + MR/1200) ........ (i)
We know,
Compound Amount when Time is T years
= P (1 + R/100) ^ T
Let us add equation (i) with the above equation:
=P (1 + R/100) ^ T + P (1 + MR/1200)
= P (1 + R/100) ^ T (1 + MR/1200) [Taking common]
Therefore,
When rate of Interest is same but Time is given in T years and M months
Compound Amount =P (1 + R/100)^T (1 + MR/1200)Compound Interest= P (1 + R/100)^T (1 + MR/1200) -P=P [(1 + R/100)^T (1 + MR/1200) -1]
Calculation of Compounded Interest half-yearly and quarter-yearly:
For calculating Compound Interest where the duration of time is less than a year, divide the given rate of interest by the frequency of time.
If duration is half-yearly, divide R% by 2, if quarter-yearly, divide R% by 2, if daily divide R% by 365.
Now, analyse how many times are you getting the Interest in one year with the given frequency of time?
If it is half-yearly, you get Interest twice a year, if quarter-yearly, you get Interest four times a year, if daily, you get Interest 365 times a year.
Now
We have;
Compound Amount = P (1 + R/100) ^ T
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In half-yearly;
Rate of Interest = R/2% = R/100*2 = R/200
Duration of Time = 2T years
So,
Compound Amount = P (1 +R/200) ^2TAnd,Compound Interest = P [(1 +R/200)^2T -1]
In quarter-yearly;
Rate of Interest = R/4% = R/100*4 = R/400
Duration of Time = 4T years
So,
Compound Amount = P (1 +R/400) ^4TAnd,Compound Interest = P [(1 +R/400)^4T -1]
For some of you, there would certainly be a question, how do know that the 'T' has to be multiplied by what number?
It is simple. You have to multiply T by the number that shows the frequency of you getting Interest in one year.
Solutions of this chapter:
- Answer: Find the rate of interest compounded yearly and the principal sum when the compound interest for 2 years and 4 years are Rs 5560 and Rs 12066.60, respectively.
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1 Comments
Solution of compound interest Class 10 - vedanta Excel in Mathematics Book is here at SciPiPupil.
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