Three natural numbers are such that one-third of the first number is equal to two-fifths of the second, and two-thirds of the second number is equal to one-seventh of the third. The sum of five times the third number and half of the square of the first number is 1348. What is the value of the second number?

Correct option is B

Given:

Let the three number is A, B , C:

​$$\frac{1}{3} A = \frac{2}{5} B$$ 

​$$\frac{2}{3} B = \frac{1}{7} C$$

Solution:

$$\frac{A}{B} = \frac{6}{5}$$

$$\frac{B}{C} = \frac{3}{14}$$​​

=> A  :    B    :    C

18x :  15x  :  70x

According to the questions:

5C + $$\frac{A^2}{2}$$ = 1348

=> 350x + 162x² = 1348

=> 81x² + 175x  - 674 = 0

=> 81x² + 337x - 162x - 674 = 0

=> x(81x + 337) - 2(81x + 337) = 0

=> (x - 2) (81x + 337)

=> x = 2 

Since, B = 15x

B = 15 $$\times$$ 2 = 30

Thus, the correct option is (b) 30.