There are three numbers If the average of any two of them is added to the third number the sums obta...........

Explanation:

 Let the three numbers be x, y, z.

(x+y)/2  + z = 177

x + y + 2z = 354 …(1)

y+z)/2 + x = 163

2x + y + z = 326 …(2)

(x+z)/2+ y = 138

x + 2y + z = 276 …(3)

Adding eq(1),(2),(3) and dividing it by 4, we get

x + y + z = 239 ,,,(4)

sub  (1) – (4),  z = 115

sub (2) – (4),   x = 87

sub (3) – (4),   y = 37

The average of largest and smallest =(115+37)/2 = 76.

Hence option 4 is the correct answer.