If at same rate of interest, in 2 years, the simple interest is Rs. 40 and compound interest is Rs. 45, then what is the principal (in Rs. )?

Correct option is A

Given:

Time = 2 years

Simple Interest (SI) = Rs. 40

Compound Interest (CI) = Rs. 45

Difference = Rs. 5

We are to find the Principal (P)

Formula Used:

The difference between CI and SI for 2 years at same rate R% is:

CI - SI = $$\frac{P \times R^2}{100^2}$$​​

Solution:​

From the formula;

​$$5 = \frac{P \times R^2}{10000} \quad \text{(1)}$$​​

Also,

SI = $$\frac{P \times R \times 2}{100} = 40 \quad \text{(2)}$$​​

From (2):

​$$\frac{2PR}{100} = 40\\ \ \\ PR = 2000 \quad \text{(3)}$$​​

Substitute PR from (3) into (1):

From (1):

​$$5 = \frac{P \times R^2}{10000} \\ \ \\5 = \frac{(PR) \times R}{10000} \\ \ \\5= \frac{2000 \times R}{10000}\\ \ \\ 5 = \frac{R}{5} \\ \ \\ \Rightarrow R = 25$$​​

Now substitute back into (3):

​$$PR = 2000\implies P = \frac{2000}{25} = Rs. \ 80$$ 

Alternate Method:

For 2 years,

​$$\text{CI} - \text{SI} = \frac{\text{SI} \times r}{100}$$​​

5 = $$\frac{40 \times r}{100}$$​​
r = 25%

P =$$\frac{40 \times 100}{25 \times 2} =$$​80