Question

a sum of money was lent for 2 yrs @20%pa. componded anually. If the intrest is payable half yearly insted of yearly, than the intrest is rs482 more. find the sum..


As got solution but still doubt---that why we have substacted amt from 1..

Answer 1

Let the sum be P.

Let the interest be compounded annually.
Accumulated total = S1 = P (1 + 20/100)² = P 1.2² = 1.44 P

       Interest I1 = accumulated sum - Principal  = S1 -  P 
                 = 1.44 P -  P = (1.44 - 1) P =   0.44 P

If the interest is compounded half-yearly, then:
         interest for 6 months = 20% * 6/12 = 10%
           number of times it is compounded is 4.

    S2 = P ( 1 + 10/100)⁴ = P 1,1⁴ = 1.4641 P
    Interest I2 = accumulated sum - Principal  = S2 -  P
                 = 1.4641 P - P  = (1.4641 - 1) P =  0.4641 P

Difference in the interest = S2 - S1 = I2 - I1 = 0.4641 P - 0.44 P 
 
      0.0241 P = Rs 482
   
         P = Sum lent = Rs 20,000.
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Shorter method:        It is not necessary to compute Interests separately.  Difference in the accumulated sums also gives difference between interests.

   Accumulated sum in case of compounding every 1 year 
           = P (1 + 20/100)² = P 1.2² = 1.44 P

   Accumulated sum in case of compounding every 6 months 
           = P (1 + 10/100)⁴ = P 1.1⁴ = 1.4641 P
 
        So  difference = 0.0241 P = Rs 482
                 P = Rs 20,000