If at same rate of interest, in 2 years, the simple interest is Rs. 40 and compound interest is Rs. 56, then what is the principal (in Rs. )?

Correct option is A

Given:

Simple Interest (SI) for 2 years = ₹40

Compound Interest (CI) for 2 years = ₹56

Rate of interest (r) is the same for both.

Formula Used:

SI =$$\frac{P \times r \times t}{100}$$​​

CI = P$$\left(1 + \frac{r}{100}\right)^t$$​- P

Solution:

( t = 2) years

SI = 40

40 =$$\frac{P \times r \times 2}{100}$$​​

40 =$$\frac{2Pr}{100}$$​​

40 =$$\frac{Pr}{50}$$​​

P r = 40$$\times$$​ 50

Pr = 2000 _____(Equation 1)

( t = 2) years

CI = 56

56 = P$$\left(1 + \frac{r}{100}\right)^2$$​ - P

56 = P$$\left[\left(1 + \frac{r}{100}\right)^2 - 1\right]$$​​

Pr = 2000

P =$$\frac{2000}{r}$$​​

56 =$$\frac{2000}{r} \left[\left(1 + \frac{r}{100}\right)^2 - 1\right]$$
Let x = $$\frac {r}{100}$$​​

56 =$$\frac{2000}{100x} \left[(1 + x)^2 - 1\right]$$​​

56 =$$\frac{20}{x} \left[(1 + x)^2 - 1\right]$$​​

56 =$$\frac{20}{x} \left[1 + 2x + x^2 - 1\right]$$​​

56 =$$\frac{20}{x} \left[2x + x^2\right]$$​​

56 = 20$$\left[2 + x\right]$$​​

56 = 40 + 20x

56 - 40 = 20x

16 = 20x

x =$$\frac{16}{20} = \frac{4}{5}$$​​

​$$\frac{r}{100} = \frac{4}{5} \implies r = 80$$​​​

Pr = 2000
P $$\times 80 = 2000$$​​

P = 25

Alternate method:  

​r =$$\frac{16}{20}$$$$\times100$$ = 80%​   

P =  $$\frac {20 \times 100}{80}$$ = 25