On a certain sum of money, the difference between the compound interest for a year payable half - yearly, and the simple interest for a year is Rs. 180. If the rate of interest in both the cases is 10%, then the sum is = ?

1Rs. 60000
2Rs. 72000
3Rs. 62000
4Rs. 54000

Answer:- 1

Explanation:-

Solution:

Rate % = 10%,
Time = 1 year
Case (I) : When interest is calculated yearly, Rate = 10%
Case (II) : When interest is calculated half - yearly

\begin{aligned} \Rightarrow New rate % = \frac{10}{2} = 5 % \\ Time  =  1 \times 2 = 2 years \\ Effective rate % \\ =  5 + 5 + \frac{5 \times 5}{100} = 10.25 % \\ Defference in rates \\ = (10.25 - 10) % = 0.25 % \\ According to the question, \\ 0 .25%  of sum  =  Rs 180 \\ Sum  = \frac{180}{0.25} \times 100 \\ = Rs 72000 \end{aligned}

$\eqalign{ & \Rightarrow {\text{New rate }}\% = \frac{{10}}{2} = 5\% \cr & {\text{Time = 1}} \times {\text{2}} = {\text{2 years}} \cr & {\text{Effective rate}}\% \cr & {\text{ = 5 + 5 + }}\frac{{5 \times 5}}{{100}} = 10.25\% \cr & {\text{Defference in rates}} \cr & {\text{ = }}\left( {10.25 - 10} \right)\% = 0.25\% \cr & {\text{According to the question,}} \cr & {\text{0}}{\text{.25% of sum = Rs 180}} \cr & {\text{Sum = }}\frac{{180}}{{0.25}} \times 100 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}72000 \cr}$