Three numbers are such that when the average of any two of them is added to the third, the results obtained are 180, 168, and 150, respectively. What is the average of the original three numbers?

Correct — A. Let the numbers be x, y, z, with sum S.

The results are (x + y)⁄2 + z, (y + z)⁄2 + x and (x + z)⁄2 + y.
Add all three: 180 + 168 + 150 = 498

The halves add to (2x + 2y + 2z)⁄2 = S
The whole terms add to x + y + z = S
So 2S = 498, and S = 249

Average = 249 ÷ 3 = 83 → option (a)

Why the others are wrong

• (b)
  84 — An average of 84 means S = 252, so the three results would total 2S = 504, not the 498 given.
• (c)
  85 — An average of 85 means S = 255, and the results would then total 510. The total of the results is always 2S.
• (d)
  86 — An average of 86 means S = 258 and a results total of 516, which is 18 more than the 498 given.

When several conditions use the same unknowns, add them all before solving any one. Each unknown appears the same number of times in the total, so the sum collapses to a multiple of x + y + z.

Here every number appears once whole and in two halves, so the three results total 2S. The individual numbers are not needed, though they come out as 111, 87 and 51.

Key facts

• (x + y)⁄2 + z = (S + z)⁄2, where S = x + y + z.
• Each third number = 2 × its result − S, which turns 180, 168 and 150 into 111, 87 and 51.
• Average of three numbers = S ÷ 3.

Common traps

• Dividing 498 by 3 and answering 166: 498 is 2S, not S.

No directly related past PYQ was found.