Question

at what rate percent a sum of rupees 1600 amount 1933 after 2 year if interest is compounded half yearly solve only with the help of logarithms g​

Answer 1

Answer:

The rate percent it takes a sum of rupees 1600 to amount to rupees 1933  after 2 years, compounded half yearly is 12%

Step-by-step explanation:

The compound interest formula is as follows:

A = P ( 1 + r/n) ⁿᵇ

Where A  = Amount accrued , Rs. 1933

          P = Principal borrowed/lend Rs. 1600

          r = Interest rate per annum ( in decimal form) , ?

          n = number of times compounded per year, (half year = 2 times in                

                 year)  = 2

          b = Number years the principal is compounded, 2 years = 2

Therefore, substitute these figures in your formula:

                      A = P ( 1 + r/n) ⁿᵇ

                   1933 = 1600 ( 1 + r/2)²ˣ²

                   1933 = 1600 ( 1 + r/2)⁴

                Divide both sides by 1600

               1933/1600 = 1600/1600( 1+r/2)⁴

   

              1.208125 = (1 + r/2)⁴

   

Introduce logarithm at this point to solve this equation, log

         log 1.208125 = log(1+r/2)⁴  

         log 1.208125 = 4log(1+0.5r).....Power rule

               0.82112 = 4 log (1+0.5r)      

Divide  both sides by 4

               0.02053 = log ( 1+ 0.5r)

This therefore means :

           10 ^ 0.0253 = (1 + 0.5r)

                1.06 = 1 + 0.5r

                0.5r = 1.06 - 1

                 0.5 r = 0.06

                      r = 0.06/0.5

                      r = 0.12

Therefore the interest rate per annum in decimal form is 0.12

Convert this to percentage = 0.12 × 100/100%

                                          = 12%

There rate percent therefore  = 12%