The average of 35 consecutive natural numbers is N. Dropping the first 10 numbers and including the next 10 numbers, the average is changed to M. If the value of M$$^2$$ - N$$^2$$ = 600, then the average of 3M and 5N is:

Given,

Average of 35 consecutive natural numbers = N

Sum of 35 consecutive natural numbers = 35N

The average after dropping the first 10 numbers and including the next 10 numbers = M

Sum of the numbers after dropping first 10 numbers and including the next 10 numbers = 35M

Every number of the next 10 numbers is 35 more than the respective number of the dropped numbers

$$=$$>  Sum increased after adding next 10 numbers = 10(35) = 350

$$=$$>  35N + 350 = 35M

$$=$$>  35M - 35N = 350

$$=$$>  M - N = 10 .................(1)

Given   M$$^2$$ - N$$^2$$ = 600

$$=$$>  (M+N)(M-N) = 600

$$=$$>  10(M+N) = 600

$$=$$>  M + N = 60 .................(2)

Solving (1) + (2)

2M = 70

M = 35

From (2), 35 + N = 60

$$=$$>  N = 25

$$\therefore\ $$Average of 3M and 5N = $$\frac{3M+5N}{2}=\frac{3\left(35\right)+5\left(25\right)}{2}=\frac{105+125}{2}=\frac{230}{2}=115$$

Hence, the correct answer is Option B