Loan & Interest

Introduction :

a) INTEREST (I): Interest is the money paid to the lender by the borrower for using his money for a specified period of the time.

b) PRINCIPAL (P): The original sum borrowed .

c) TIME (n): Time for which money is borrowed

d) RATE OF INTEREST (r) : Rate at which interest is calculated on the original sum .

e) AMOUNT (A): Sum of Principal
and Interest. ** A = P + I **

1) SIMPLE INTEREST

When the interest is calculated every year (or every time period) on the original principal i.e. , the sum at the beginning of first year , such interest is called SIMPLE INTEREST

Using above formula, Value for Principal, Rate and Time can be derived as:

As we know from above, Amount is Sum of Principal and Interest:

Example 1: If Rs 4000 becomes Rs 4800 in 2 years , what will Rs 6000 become at the end of 6 years at the same rate of interest under simple interest ?

Principal (P) = Rs 4000, Amount (A) = Rs 4800 & Time (T) = 2 Years

So, I = 4800 - 4000 = Rs 800

R = 10 %, P = 6000, Time = 6 Years

Since, Amount = Principal + Interest

Amount = 6000 + 3600 = 9600 Rs

**Note 1 :**

If a sum of money becomes n times in T years at simple interest , then the rate of Interest (I) is given by :

Example 2: Find the rate of Interest if sum of money becomes 3 times in 5 years ?

**Note 2 :**

If a certain sum of money is lent out in n parts in such a manner that equal sum of money is obtained at simple interest on each part where interest rates are R1, R2, ... , Rn respectively and time periods are T1, T2, ... , Tn respectively, then the ratio in which the sum will be divided in n parts can be given by

Example 3: If a sum of Rs 1600 is divided into two such parts that the simple interest on the first part for 2(1/2) years at the rate of 4% p.a. equals the simple interest on the second part for 5 years at the rate of 3% p.a. , then find two such divisions of the sum ?

Given: R1 = 4% p.a., T1 = 2(1/2) = 5/2 Years , R2 = 3 % p.a. and T2= 5 Years

Using above formula:

Sum of proportional = 3 + 2 = 5

Therefore , 1st Part = 3/5 * 1600 = Rs 96

And , 2nd part = 2/5 * 1600 = Rs 640

Alternate: You can check answers given by Trial and Error i.e. Test given answers until the right one is found.

**Note 3 :**

If a certain sum of money becomes n times itself in T years at a simple Interest , then the time T in which it will become m times itself is given by :

Example 5: A sum of money put on simple interest doubles itself in 12 (1/2 ) years In how many years would it treble itself ?

Given : n = 2 , m = 3 and T = 12 (1/2 ) = 25/2 years

**Note 4 :**

If Rate of Interest ( R ) and Principal (P) are given and need to find the time period in which the principal gets n times , then find time period using below :

Example 6: In how many years a sum of Rs 5000 at simple interest of 10% will become double , triple and 4 times ?

Given : Principal (P) = Rs 5000 & Simple Interest (SI) = 10% .

Using above formula :

2) Compound Interest

Under Compound Interest , the Interest is added to the principal at the end of each period to arrive at the new principal for the next period.

In other words , the amount at the end of first year (or period) will become the principal for the second year (or period) ; the amount at the end of second year (or period) becomes the principal for the third year (or period) and so on.

a)Interest Compounded Annualy :

The Amount A due after t years , when a principal P is given on
compound interest at the rate R % ** per annum ** is given by :

A is the Amount

P is the principal amount

R is the rate percentage

n is the time period

As we know, Amount (A) is the Sum of the Principal (P) and the Compound Interest (I):

** A = P + I **

** I = A - P **

Note : The Compound Interest for the first year (where compounding is done every year) is the same as the Simple Interest for one year.

Example 7: A sum of Rs 1000 at 4% p.a. would amount to Rs ---- after 2 years by Compound Interest ?

Given : P (Principal) = Rs 1000 , R (Rate of Interest)= 4% , Time Period (n) = 2

Note : Compound interest can be calculated annually, half-yearly, quarterly or even daily.

b) Interest Compounded Half - Yearly :

The Amount A due after t years , when a principal P is given on
compound interest at the rate R % ** per half-annum ** is given
by :

Example 8: Find the amount of Rs 8000 in one and half years at 5% per annum compound interest payable half-yearly ?

Given : P = 8000 , R = 5 and t = One and half year = 3/2

c) Interest Compounded Quaterly :

The Amount A due after t years , when a principal P is given on
compound interest at the rate R % ** per quarter ** is given by
:

Example 9: Find the compound interest on Rs 4000 at 24% per annum for 3 months , compounded monthly ?

Given : P = Rs 4000 , R = 24% and n = 3 month = 3/12 year

**Note 5 :**

When the rates of Interest are different for different years , say R1 %, R2 % , R3 % for first , second and third year respectively , then

Example 10: : Ram invests Rs 5000 in a bond which gives interest at 4 % per annum during the first year , 5 % during the second year and 10% during the third year . How much does he get at the end of the third year ?

Given : P = Rs 5000 , R1 = 4 % , R2 = 5% and R3 = 10%