Three numbers are in the ratio $$\frac45: \frac56: \frac9{10}$$ . The difference between the smallest and the greatest numbers is 12. Find the number which is NEITHER the smallest NOR the greatest.

Correct option is C

Given:
Three numbers are in the ratio

​$$\frac45: \frac56: \frac9{10}$$​​

Solution:
=> 4/5 × 12 = 9.6
=> 5/6 × 12 = 10
=> 9/10 × 12 = 10.8
The three numbers are in the ratio = 9.6:10:10.8
Let the numbers be 9.6x, 10x, and 10.8x.
=> 10.8x - 9.6x = 12
=> 1.2x = 12
=> x = 10
The numbers are 96, 100 and 108
The number which is NEITHER the smallest NOR the greatest is 100.

Alternate Method

LCM of 5, 6 and 10 is 30.

4/5 × 30 = 24

5/6 × 30 = 25

9/10 × 30 = 27

=> A ∶ B ∶ C = 24x ∶ 25x ∶ 27x

According to the question

=> 27x - 24x = 12

=> 3x = 12

=> x = 4

1st number = 24x = 24 × 4 = 96

2nd number = 25x = 25 × 4 = 100

3rd number = 27x = 27 × 4 = 108
The number which is NEITHER the smallest NOR the greatest is 100.