The average of 13 numbers is 70. The average of the first five numbers is 60. The average of the next five numbers is 25 % higher than the average of the first five numbers. The 11th number is 6 less than the 13th number, and the 12th number is 4 greater than the 13th number. What is the average of the 11th and 12th numbers?

Correct — A. Work with totals: total = average × count.

All 13: 13 × 70 = 910
First five: 5 × 60 = 300
Next five: average 60 × 1.25 = 75, so total 5 × 75 = 375
Last three: 910 − 300 − 375 = 235

Let the 13th be x, so the 11th is x − 6 and the 12th is x + 4
Sum: (x − 6) + (x + 4) + x = 3x − 2 = 235, so x = 79
11th = 73, 12th = 83
Average of the 11th and 12th: (73 + 83) ÷ 2 = 78 → option (a)

Why the others are wrong

• (b)
  77 — The pair averages 1 less than the 13th, so 77 makes the 13th 78. Then the last three are 72 + 82 + 78 = 232, not the 235 left over.
• (c)
  76 — It makes the 13th 77, so the last three are 71 + 81 + 77 = 229. The totals leave 235 for them.
• (d)
  74 — It makes the 13th 75, and 69 + 79 + 75 = 223, well short of the 235 left after the first ten numbers.

With several group averages, turn each into a total (average × count), subtract to find what is left, and only then bring in the individual numbers.

Taking the 13th number as x keeps the link simple: the 11th and 12th average to (x − 6 + x + 4) ÷ 2 = x − 1, so once x = 79 the answer is 78 without finding each number.

Key facts

• Total = average × count: 13 × 70 = 910.
• 25% higher than 60 is 60 × 1.25 = 75, not 60 + 25.
• If a = x − 6 and b = x + 4, then (a + b) ÷ 2 = x − 1.

Common traps

• Adding 25 to 60 instead of raising it by 25%: that makes the next five total 425 and leaves only 185 for the last three.
• Stopping at the 13th number (79) or the average of all three (235 ÷ 3 ≈ 78.33) instead of the average of the 11th and 12th.

No directly related past PYQ was found.