Simple and Compound Interest

                         Simple & Compound Interest

If I borrow a certain sum of money for a certain period from a money lender, I am expected to pay a certain extra sum of money at a fixed rate for the use of the money borrowed. The extra sum thus paid is called Interest. The money borrowed is named the Principal and the sum of interest and principal together is called the AMOUNT.

The interest expressed as a percentage of the principal for a period of one year is called the Rate percent per annum. The words per annum are generally omitted. When we say "Rate = 5%", we mean 5% per annum.

Simple Interest (S.I.)

When the interest is paid as it falls due, i.e. at the end of every period (e.g. yearly, half yearly or quarterly), the principal is said to be lent or borrowed at Simple Interest.

If        P = Principal,                  R = Rate per annum,

T = Time in years,          SI = Simple interest,

A = Amount

                

   SI   =  P x R x T

                  

                  100

A  =  P + SI  =  P[1+RT/100 ]

E16.If Rs.650 amounts to Rs.790 in 4 years, then what sum of money will it amount to in 7 years at the same rate of interest?

Sol. S.I. = Rs.(790 – 650) = Rs.140. Also, 

SI   =  P x R x T

                  

               100

=140  =   650 x (R) x (4)  =  R  =              140 x 100   =   70                 

                           100                                        650x 4     13

Amount = P[1+RT/100 ]= 650(1 + 70 x 7/ 13x100)

= Rs.895.

short-cut: You can see that Rs.140 is earned in 4 years

Rs.35 is earned in 1 year = Rs 35 x 7

= Rs.245 is is earned in 7 years etc.

Compound Interest (C.I.)

When the interest, as it becomes due, is added to the principal, and the interest for the next period is calculated on the new principal, then the money is said to be lent or borrowed at Compound Interest.

=        First year's principal + First year's Interest

   = Second year's principal.

If P = Principal, A = Amount in n years, R = Rate of interest per annum, then

A = P(1+R/ 100]ⁿ interest payable annually

A =  P(1+R’/ 100]^n’, interest payable half-yearly

    R'= R/2, n'= 2n

A = P(1 +R/400)⁴ⁿ , interest payable quarterly

 

           1  +   R      is the yearly growth factor;

  100

 

           1 –    R       is the yearly decay factor or depreciation factor.

                   100              

Points to remember

A.         When time is fraction of a year, say 4¾, years, then,

 Amount = P  1 + R   ⁴  x   1+ ¾ R

                           100               100

B.         CI = Amount– Principal = P( (1 + R /100)ⁿ – 1 )

C.        When Rates are different for different years, say R₁, R₂, R₃% for I^st ,  2^nd   &   3^rd years respectively, then,

 Amount = P[1 +R₁ /100 ][1 +R₂/1 00 ][1 + R₃/100]

In general, interest is considered to be SIMPLE unless  otherwise stated.

E17.A certain sum of money at C.I. amounts to Rs.811.25 in 2 years and to Rs.843.65 in 3 years. Find the sum of money.

Sol. Since A = P(1+R/ 100]ⁿ

811.25 = P[1+  R/100]²       ...(1) and

843.65=P[1+R/100]³              ...(2)

On dividing (2) by (1), we get: 843.65 =   1 + R

                          811.25        100

=1.04=1+ R = R = 4

  100

Now, putting R = 4 into (1), we get

811.25=P(1+4/100)² =P=750

The sum of money is Rs.750.

Equal annual instalment to pay the debt (Borrowed) amount

Let the value of each equal annual instalment = Rs.a. Rate of interest = R% p.a.

Number of installments per year = n.

Number of years = T.

.•. Total number of installments = n x T.

Borrowed amount = B. Then,

 a         100     +        100       ² + ……… +      100    ^nT           =   B

         100 + R        100 + R                          100+R

E19.A loan of Rs.2000 is to be paid back in 3 equal annual installments. How much is each 

installment to the nearest whole rupee, if the interest is compounded annually at 12.5% p.a.?

Sol. 2000 = a{(100/112.5) + (100/112.5)² + (100/112.5)³} = Rs.840 (approx)