The difference between the compound interest and the simple interest on a certain sum at 10 % per annum for 2 years is ₹ 52 . The sum is :

Correct option is B

Given:
The difference between the compound interest and the simple interest on a certain sum at 10% per annum for 2 years is 52.
Formula Used:
Simple Interest (SI) = $$\frac{P \times R \times T}{100}$$

$$A = P(1 + \frac{r}{100})^n$$

$$A = P(1 + \frac{r}{100})^n - P$$

​​​​Where:
A is amount,
P is the principal amount (the initial amount of money)
R is the rate of interest per period (as a percentage)
T is the time the money is invested or borrowed for, in years

Solution:
For 2 years,

​$$52 = \left[P \left(1 + \frac{10}{100}\right)^2 - P \right]- \frac{P \times 10 \times 2}{100}$$

​​$$52 =\left[P \left(\frac{110}{100}\right)^2 - P \right]- \frac{P \times 20}{100}$$

$$52 = [P \left(1.1)^2 - P \right]- 0.2P$$

$$52 = [1.21P - P ]- 0.2P$$

52 = 0.21P - 0.2P
52 = 0.01P

P = $$\frac{52}{0.01}$$

P = $$\frac{5200}{1}$$

P = 5200
So, the sum is 5200.
Thus, the correct answer is (b).
Alternative Method:
Let the principle is 100x.​​

Simple in 1st year = $$100x \times \frac{10}{100}$$ = 10x

Simple in 2nd year = $$100x \times \frac{10}{100}$$ = 10x

Total simple interest = 20x

Total rate of compound interest = $$10\% + 10\% + \frac{10\times 10}{100}$$ = $$10\% + 10\% + 1\% = 21\%$$

Compound interest = $$100x \times \frac{21}{100}$$= 21x

21x - 20x = 52
x = 52
Principle = 100x = 100×​ 52 = 5200​​​