The average of 15 numbers is 80. The average of the first 6 numbers is 72. The average of the next 6 numbers is 25% more than the average of the first 6 numbers. The 13th number is 8 more than the 15th number, and the 14th number is 10 less than the 15th number. What is the average of the 13th and 14th numbers?

Correct — C. Turn averages into totals, then isolate the last three.

All 15: 15 × 80 = 1200
First 6: 6 × 72 = 432
Next 6: average 72 × 1.25 = 90, sum 6 × 90 = 540
Last three: 1200 − 432 − 540 = 228

Let 15th = x, so 13th = x + 8 and 14th = x − 10
Add them: 3x − 2 = 228 → x = 230⁄3 ≈ 76.67
Average of 13th and 14th = (2x − 2) ÷ 2 = x − 1
= 76.67 − 1 = 75.67 → option (c)

Why the others are wrong

• (a)
  70.89 — The answer equals x − 1, so 70.89 makes the 15th number 71.89 and the last three 3 × 71.89 − 2 = 213.67, not 228.
• (b)
  85 — 85 makes the 15th number 86 and the last three 3 × 86 − 2 = 256, which overshoots the 228 left after the first twelve.
• (d)
  80.65 — 80.65 makes the 15th number 81.65 and the last three 3 × 81.65 − 2 = 242.95, still above the 228 that is left.

Average problems open up once every average becomes a total: sum = average × count.

Subtract the known blocks from the grand total and what is left belongs to the unknown numbers. Tie them all to one variable, here the 15th number, and a single equation finishes the job.

The shortcut: x + 8 and x − 10 average to x − 1, so the answer is one less than the 15th number.

Key facts

• Sum = average × count, so 15 numbers averaging 80 total 1200.
• 25% more than 72 is 72 × 1.25 = 90.
• Two numbers x + 8 and x − 10 average to x − 1.

Common traps

• Reading '25% more' as 72 + 25 = 97 instead of 72 × 1.25 = 90.
• Stopping at x = 76.67, the 15th number, instead of the average of the 13th and 14th.

No directly related past PYQ was found.