Three numbers are such that when the sum of any two of them is added to half of the third, the results are 210, 198, and 192 respectively. What is the average of the three numbers?

Correct — A. Add the three results and count how often each number appears.

Let the sum of the three numbers be S.
Add the results: 210 + 198 + 192 = 600

Each number is counted in full twice and halved once: 1 + 1 + ½ = 2½ times
So 2½ × S = 600, and S = 600 ÷ 2.5 = 240

Average: 240 ÷ 3 = 80 → option (a)

Why the others are wrong

• (b)
  70 — An average of 70 means a sum of 210, and 2½ × 210 = 525, not 600. The three results add to 600, so the sum must be 240.
• (c)
  60 — 60 is one of the three numbers, not their average. It is the number halved in the 210 result: 240 − 60⁄2 = 210. The remaining two are 84 and 96.
• (d)
  50 — An average of 50 means a sum of 150, and 2½ × 150 = 375, far below the 600 that the three results add to.

When every condition involves all the unknowns, add the conditions. Each result here is two numbers in full plus half the third, so the grand total counts each number 2½ times.

That turns three equations into one, 2½S = 600. The individual numbers are not needed for the average.

Key facts

• Adding symmetric conditions multiplies the sum S by the weight each number carries.
• Here the weight is 1 + 1 + ½ = 2½, so S = 600 ÷ 2.5 = 240.
• Sum of two numbers + half the third = S − half the third.

Common traps

• Counting each number twice and getting S = 300. The half of the third adds another ½, making 2½.
• Solving for one number, such as 60, and stopping before taking the average.

No directly related past PYQ was found.