In a set of four consecutive multiples of 3, where the multiples are ordered from least to greatest, the first...

1. TRANSLATE the given information about consecutive multiples

• Given information:
  • Four consecutive multiples of 3, ordered from least to greatest
  • First multiple is represented by y

• What this tells us: If the first multiple of 3 is \(\mathrm{y}\), then consecutive multiples of 3 differ by 3 each time:
  • First multiple: \(\mathrm{y}\)
  • Second multiple: \(\mathrm{y + 3}\)
  • Third multiple: \(\mathrm{y + 6}\)
  • Fourth multiple: \(\mathrm{y + 9}\)

2. TRANSLATE the relationship described in words

• The problem states: "The product of 8 and the second multiple is at most 18 less than the sum of the first and fourth multiples"

• Breaking this down piece by piece:
  • "Product of 8 and the second multiple" → \(\mathrm{8(y + 3)}\)
  • "Sum of the first and fourth multiples" → \(\mathrm{y + (y + 9)}\)
  • "At most 18 less than [something]" → \(\leq\) [something] - 18

3. INFER the correct inequality structure

• "At most" means \(\leq\) (less than or equal to)
• "18 less than the sum" means we subtract 18 from the sum
• So: \(\mathrm{8(y + 3) \leq y + (y + 9) - 18}\)

4. Match with answer choices

Looking at the options, this matches choice (A) exactly.

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "at most 18 less than" and think it means the sum should be at most 18 less than the product, reversing the inequality.

This leads them to write: \(\mathrm{y + (y + 9) \leq 8(y + 3) - 18}\), which when rearranged becomes \(\mathrm{8(y + 3) \geq 18 + y + (y + 9)}\). Looking for something similar, they might select Choice B: \(\mathrm{8(y + 3) \geq 18 - (y + (y + 9))}\) even though the algebra isn't quite right.

Second Most Common Error:

Poor INFER reasoning: Students incorrectly determine the consecutive multiples, thinking the second multiple is \(\mathrm{y + 6}\) instead of \(\mathrm{y + 3}\), perhaps by confusing "second multiple" with "second gap" or miscounting.

This leads them to use \(\mathrm{8(y + 6)}\) in their inequality, making them select Choice C: \(\mathrm{8(y + 6) \leq y + (y + 3) - 18}\) or Choice D: \(\mathrm{8(y + 6) \geq 18 - (y + (y + 3))}\).

The Bottom Line:

This problem tests your ability to carefully translate complex English phrases into mathematical symbols while keeping track of multiple pieces of information. The phrase "at most 18 less than" is particularly tricky because it requires understanding both the inequality direction and the algebraic structure simultaneously.