Compound Interest Formula- Definition, Derivation, Some examples

To Learn the Compound Interest formula from our best teachers. You can find the definition, formula, derivation, calculation & solved examples.

At the end of the first years ,if the Interest accrued is not paid to the moneylender , but is added to the principal, then this amount becomes the principal for the next year and so on. This process is repeated until the amount for the whole time is found.

The difference between the final amount and the original principal is called the Compound interest.

In mathematics, compound interest usually denoted by C.I. It generally used in most of the transactions in the banking and finance sectors and other areas.

Compound Interest Formula:

Compound interest (C.I.) can easily calculated by the following formula;

        Formula : A = P ( 1 + r/100)^nt 

Here, A = Final amount 

          P = Principal 

          r = rate of interest compounded yearly. 

          t = Time ( in years ) 

          n = Number of times interest is compounded per year. 

 We can write the alternative formula as given below;

 COMPOUND INTEREST = AMOUNT- PRINCIPAL. 

C.I.=A – P 

Similarly, 

        We know, 

              [ A=P(1+r/100)^nt ] 

Therefore, C.I.= P(1+r/100)^nt -P 

C.I.= P[(1+r/100)^nt – 1]

This formula of compound interest is also known as the periodic compounding formula. 

Here, 

Principle =The money borrowed ( or the money lent )is called the Principle.

Amount =The sum of the Principal and the interest is called the Amount.

Thus, 

Amount = principal + interest.

Rate = It is the interest paid on rs. 100 for a specified period. 

Time = It is the time for which the money is borrowed. 

Number = It is the compounding frequency or the number of times interest is compound in a year.

Compound Interest Formula for Different Time Periods

In case of compound interest, the values of amount and interest for different years are given below;

Compounded Annually Formula	A = P (1 + r)t
Compounded Semi-Annually Formula	A = P (1 + r/2)2t
Compounded Quarterly Formula	A = P (1 + r/4)4t
Compounded Monthly Formula	A = P (1 + r/12)12t
Compounded Weekly Formula	A = P (1 + r/52)52t
Compounded Daily Formula	A = P (1 + r/365)365t

This chart formula are help to determine the interest and amount in case of compound interest quickly.

Compound interest when the rate is compounded half- yearly.

The time from one specified interest period to the next period is called a Conversion Period. If this specified period is six months, then the interest is compounded Semi-annually or Half-Yearly, then there are two Conversion periods in a year.  

In this case , we can calculate the compound interest on a principal, P for 1 year at an interest rate R% compound half-yearly.

   Let,  

          S.I1 = P× R× 1/100× 2 

Amount at the end of first six months,

      A1 =P+ S.I.

            = P + P×R×1/ 2× 100 

            = P( 1+ R/ 2× 100) 

Therefore, A1 = P2 

Simple interest for next six months, now the principal amount has changed to P2 – 

        S.I.2 = P2× R×1 /100×2 

Amount at the end of 1 year,

A2 =P2 + S.I.2 

       = P2 + P2 × R× T / 100× 2

or,  A2 = P2 ( 1+ R/ 2× 100)             

       = P(1+ R/2×100)×P(1+ R/ 2×100)         

       =P( 1 + R/ 2× 100)²

Now we have the final amount at the end of 1 year:

A = P ( 1 + R /100×2 )²

Rearranging the above equation,

        A = P( 1 + R/2 / 100)²

Let,  

        R/2 = R’ 

        2T =T’  

the above equation can be written as, [for the above case] 

      T = 1 Year

     A=  P ( 1 + R’ / 100) ^T’

Hence, when the rate compounded half-yearly, we divide the rate by 2 and multiply the time by 2 before using the general formula for compound interest.

Quarterly Compound Interest Formula

The time from one specified interest period to the next period is called a Conversion Period. If this specified period is three months, then there are four Conversion Periods in a year. 

In this case, we can calculate the compound interest on a principal, P kept for 1 year at an interest rate R% compounded quarterly.

Thus, we can give the interest compounded quarterly formula as given below- 

A = P (1+ R/4/100) ^4T 

C.I.= A-P 

or ,C.I.= P( 1 + R/4/100 )^4T -P   

Here,

       A= amount 

       P= Principal 

       R= rate of per year per year 

       T= Number of years 

Derivation of compound interest formula

C.I.= P[ ( 1 + R/100 )^n -1] 

Let, 

      Principal =P 

      Time = n years 

      Rate = R 

S.I for the first years 

S.I1 = P×R×T/ 100 

Amount after first year =P+S.I1 

                                           =P+P×R×T/100

                                           =P ( 1 + R/ 100) 

                A= P2 

Simple Interest (SI) for second year:

S.I.2= P2×R×T/100 

Amount after second year =P2 +S.I2 

                          = P2 + P2×R×T/ 100

                          = P2 (1+R/100)    = P(1+R/100)×(1+R/100)

    A = P( 1 + R/100)²

Similarly, if we proceed further to n

years, we can deduce:

           A = P( 1+R/100)^n 

        C.I.=A – P

              =P(1 + R/100) – P 

       C.I =P[(1+ R/100) -1]  (PROVED)

REMEMBER SOME POINTS WHEN SOLVING PROBLEMS ON COMPOUND INTEREST:

 1.   A=P(1+R/100)^n.  and 

       C.I. =P[ (1+R/100)^n -1 ] 

Here,

       A= amount 

       P= Principal 

       R= rate of per year per year 

       T= Number of years 

2. When the rate of interest for the successive fixed periods are r1%, r2%, r3%,…., then amount A given by –

A = P(1+r1/100).(1+r2/100).(1+r3/100)…….

3. S.I and C.I are equal for the first time conversion period on the same sum and at the same rate. 

4. C.I of 2nd conversion period is more than 

C.I of 1st conversion period and C.I of 2nd conversion period –  C.I of 1st conversion period = S.I on the interest of the 1st conversion period.

5. C.I for the nth conversion period = amount after n conversion – amount after (n-1) conversion period.

6. When the total time is not a complete number of conversion period, we consider simple interest for the last partial period. For example, if time is 2 years 5 month and the interest is r% per annum compounded annually, then

A=P(1+r/100)^2 * ( 1+ 5r/12 /100)

Solved Examples of Compound Interest

Example 5: What is the compound interest on 20000 for one year at the rate of 20% per annum, if the interest is compounded quarterly?

Solution: Given,

Principal P = Rs 20000

rate R = 12% (12/4 = 3 % per quarter year)

Time = 1 year (1 × 4 = 4 quarters)

by formula,

A = P (1 + R/100)ⁿ

= 20000 (1 + 3/100)⁴

= 20000 (103/100)⁴

Therefore, A  = 22510.17

Compound Interest = A – P

= 22510.17– 20000

= 2510.17

Therefore, Compound Interest = 2510.17 (Answer)

Example 6: Find the compound interest at the rate of 5% per annum for 2 years on that principal which in 2 years at the rate of 4% per annum given Rs. 200 as simple interest.

Solution: Given,

Simple interest SI = 200

rate R = 5%

time T = 2 years

by formula,

Simple interest = (P × T × R)/100

P = (SI × 100)/T × R

= (200 × 100)/2 × 5

= 20000/10

Therefore, P = Rs 2000

Rate of Compound Interest = 4%

time = 2 years

by formula ,

A = P (1 + R/100)

= 2000 (1 + 4/100)

= 2080

Compound Interest = A – P

= 2080– 2000

= 80

Therefore, Compound Interest = 80 (Answer)

Practice Questions

• Find the Compound Interest when principal = Rs 5000, rate = 7% per annum and time = 3 years?
• Find the compound interest at the rate of 3% per annum for 1 years on that principal which in 3 years at the rate of 5% per annum given Rs. 500 as simple interest.
• What will be the compound interest on Rs 5000 in three years when the rate of interest is 4% per annum?
• Jyoti deposited Rs. 5000 with a finance company for 3 years at an interest of 3% per annum. What is the compound interest that Shreya gets after 3 years?
• What is the compound interest on 8000 for one year at the rate of 10% per annum, if the interest is compounded quarterly?

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