Three numbers are in the ratio $$\frac{1}{2} : \frac{2}{3} : \frac{3}{4}$$. The difference between the greatest and the smallest number is 27. The average of the three numbers is:

We have numbers as :
$$\frac{1}{2}:\frac{2}{3}:\frac{3}{4}$$
Now making the denominator common which will be the LCM of 2,3 and 4
we get :$$\frac{6}{12}:\frac{8}{12}:\frac{9}{12}$$
so we can say the numbers are in the ratio 6:8:9
Now let the common factor be x
so we get numbers as 6x,8x and 9x
Now as per given condition

difference between the largest and smallest number is 27.

i.e, 9x - 6x = 27

we get 3x=27
x=9
so numbers are 54 ,72 and 81
Now the average of numbers = $$\frac{\left((54+72+81\right)}{3}=69$$

Hence , option C is correct.