What will be the sum of two numbers?
I. Among the two numbers, the bigger number is greater than the smaller number by 6.
II. 40 % of the smaller number is equal to 30 % of the bigger number.
III. The ratio of half of the bigger number to one third of the smaller number is 2 : 1.

Statement 1: Let the smaller number be x, then the bigger number becomes x + 6.

Statement 2: Let the smaller number be a and the bigger number be b. Given that 40% of a is 30% of b,

$$\frac{40}{100}\ \times\ a\ =\ \frac{30}{100}\ \times\ b$$

$$\frac{a}{b}\ =\ \frac{3}{4}\ $$

Statement 3: Let the smaller number be a and the bigger number be b. The ratio of half of the bigger number to one-third of the smaller number is 2: 1, which means,

$$\dfrac{\frac{b}{2}}{\frac{a}{3}}\ =\ \frac{2}{1}$$

$$\frac{3b}{2a}\ =\ \frac{2}{1}$$

$$\frac{a}{b}\ =\ \frac{3}{4}$$

Case 1 : I and II

Let smaller number be  x => bigger number = x+6

$$\frac{40}{100}\cdot x=\frac{30}{100}\left(x+6\right)$$

=> x=18 , Sum of bigger and smaller number = 42

Case 2 : I and III

Let smaller number be x => bigger number = x+6

$$\frac{\left(x+6\right)}{2}\cdot\frac{3}{x}=\frac{2}{1}$$

On solving, x=18. So sum = 42

Case 3 : II and III

The conclusion from both statements is the same, and the sum of the numbers cannot be obtained using both statements.

Hence, the correct answer is option C.