The sum of two consecutive even numbers is 174. Find the smaller number.

Correct — A. Consecutive even numbers differ by 2, so call them n and n + 2.

n + (n + 2) = 174
2n + 2 = 174
2n = 172 → n = 86

The pair is 86 and 88. The question asks for the smaller, which is 86 → option (a).

Why the others are wrong

• (b)
  84 — 84 would pair with 86, and 84 + 86 = 170 — four short of 174.
• (c)
  88 — 88 is the larger member of the correct pair. It does sum to 174 with 86, but the question asks for the smaller number, and this is the trap the option exists for.
• (d)
  90 — 90 would pair with 92 for a sum of 182, eight too many. Every step of 2 in the smaller number moves the sum by 4.

For any short run of consecutive numbers the mean sits at the middle of the run.

Two consecutive even numbers average 174 ⁄ 2 = 87, so they lie one on each side of 87: 86 and 88. No algebra is needed once you see that.

Written out, n and n + 2 give 2n + 2 = S, so the smaller is (S − 2) ⁄ 2. The same idea scales: three consecutive even numbers summing to S have middle term S ⁄ 3.

Key facts

• Two consecutive even numbers summing to S are (S − 2) ⁄ 2 and (S + 2) ⁄ 2.
• Their average is S ⁄ 2, here 87, the odd number lying between them.
• The sum of two consecutive even numbers is always even but never a multiple of 4.
• Consecutive integers step by 1, while consecutive even or odd numbers step by 2.

Common traps

• Answering 88, the larger of the correct pair, when the smaller was asked for.
• Setting up n and n + 1, which describes consecutive integers rather than consecutive even numbers.
• Dividing 174 by 2 and answering 87, which is odd and so cannot be either number.

No directly related past PYQ was found.