Simple and Compound Interest for Quantitative Techniques

Simple and Compound Interest​​ for Quantitative Techniques

  • Principal:​​ The​​ Money lent or borrowed for a​​ particular​​ period​​ is called the Principal, also known as the Sum of money. Denoted usually by P.

  • Interest:​​ When a borrower pays extra money to the lender to compensate for the opportunity cost of the lender, is called the Interest. It is usually represented in the form of percentages. Usually denoted by R. It can be of two different kinds: Simple or Compound Interest.

  • Simple Interest:​​ The interest is fixed over the period and doesn’t interest on itself.

Let the time be denoted by T.​​ 

Simple Interest=P*R*T100

Amount = S.I. + P

 

Example:​​ Find the simple interest on Rs. 60000 at 6.5% per annum for 4.5​​ yrs.

Solution:​​ P = 60000 ; R = 6.5% ; T= 4.5​​ yrs

  ​​ ​​ ​​​​ S.I = P*R*T/100 = 60000*4.5*6.5/100 = 6*45*65 = Rs. 17550

  • If the​​ Amount equals to double the Principal​​ at simple interest​​ in T yrs, then​​ R = ( 100/T ) %

Example:​​ If the Principal amount is Rs.100 which amounted to Rs. 200 in 3 yrs, then find the Rate of Interest.

Solution:​​ The condition given is of double Principle hence we can use the above formula

  ​​ ​​ ​​​​ R = (100/3) =33.33%

Alternate: S.I = 200 - 100​​ 

 ​​ ​​​​ 100=100(R*3/100)

  ​​ ​​​​ 1 = ​​ R*3/100

  ​​ ​​ ​​​​ R = 100/3 = 33.33%

  • If the​​ Amount equals to triple the Principal​​ amount in T yrs at R%, then​​ R = (200/T) %.

  • If the​​ Amount equals to​​ n times​​ the Principal​​ amount in T yrs at R%, then​​ R =((n-1)*100/T) %

  • Compound Interest:​​ Concept of interest on interest is applicable in case of compound interest. Hence the lender earns interest on completion of one time period and further interest on the earlier earned interest.

 

When​​ money is​​ compounded Annually:

Compound Amount=P1+R100T

 

The compound interest formula can be varied for the different time by substituting R by R/n and T by T*n where n is how frequent compounding is being done.

 

For Example:

 

When​​ money is​​ compounded Semi-Annually:

 

Compound Amount=P1+R2*100T*2

 

When​​ money is​​ compounded​​ Quarterly:

 

Compound Amount=P1+R4*100T*4

 

When​​ money is​​ compounded Monthly:

 

Compound Amount=P1+R12*100T*12

 

Compound Interest = Compound Amount - Principal

 

Compound Interest=P1+R100T-P

 

C.I.=P1+R100T-1

 

 

Example: Find the compound interest on Rs. 10000 at 5% per annum for 2 yrs, compounded annually.

 

Solution:​​ C.I ​​ = P[​​ (1+R/100)2​​ -1] = 10000​​ [(1+5/100)2-1]​​ 

​​ = 10000​​ [(21/20)2​​ -1] = 500

 

Example: If I invest Rs. 2000 for 3 yrs at 25% p.a. and earn compound interest. Which compounding will yield me​​ a​​ better​​ amount​​ yearly or half-yearly.

 

Solution:​​ Principal = Rs. 2000, Time =3 yrs and Rate of interest= 25%

  ​​ ​​ ​​​​ C.Amount​​ (yearly) = P​​ (1+R/100)^T = 2000​​ (1+​​ 25/100)3

  ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​​​ = 2000​​ (5/4)3​​ = 2000​​ *​​ 1.95 = 3906.25

  ​​ ​​ ​​​​ C.Amount​​ (half-yearly) = P​​ (1+​​ R/2*100)^T*2 = 2000​​ (1+​​ 25/200)3*2

  ​​ ​​​​  ​​ = 2000​​ (9/8)6​​ = 4054.57

 

 

Compounding done semi annualy will amount more in the given investment.​​ 

 

  • When Rates are different for different years lets say R1, R2 and R3 for three consecutive years ​​ then,

Compound Amount=P1+R11001+R21001+R3100

 

  • When the Principal becomes x in T years, then the rate of the compound is

 

R=100x1T-1

 

  • When the Principal amount A1 in T years ​​ while it amounts to A2 in (T+1) years at CI, Then

R=A2-A1A1*100

  • If a person takes a loan from the bank at R% and agrees to pay it back in equal instalments for T years, then the value of each instalment is:

 

P=X1+R100TX1+R1002+X1+R1001

 

Where X is the instalment amount.

 

  • Rule of 72:​​ If the amount equals to double the Principal at R% in T yrs then we can use the formula​​ R = 72/T or T = 72 / R​​ to find either R and T.​​ 

  • Suppose C.I of 7th​​ year = P(1+R/100)7​​ - P(1+R/100)6

​​ = P(1+R/100)6​​ *R/100

  • Hence using the above example we can say​​ 

 

Compound Interest on nth year:

​​ ​​ P1+R100T-1*R100

 

  • Difference between Compound Interest and Simple Interest ​​ if ​​ calculated for the same number of years and Same rate of interest

  • T = 1yr;​​ 

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

S.I = PR/100

Difference =C.I – S.I = PR/100 – PR/100 = 0

  • T= 2 yrs;

C.I = P(1+R/100)2​​ – P

S.I = 2PR / 100

C.I – S.I = [P(1+R/100)2​​ – P] - 2PR / 100

​​ = PR2/1002​​ (on solving previous equation)

  • T = 3 yrs;

C.I = P(1+R/100)3​​ – P

S.I = 3PR / 100

C.I – S.I = [P(1+R/100)3​​ – P] - 3PR / 100

​​ = PR2/1002​​ [R/100 +3]

  • T = n yrs;

C.I – S.I = (C.I – S.I)2nd year​​ * (R/100 + 3)

 

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