The number y is 84 less than the number x. Which equation represents the relationship between x and y?

1. TRANSLATE the problem information

• Given information:
  • The number \(\mathrm{y}\) is 84 less than the number \(\mathrm{x}\)
  • Need to find the equation representing this relationship

• What this tells us: I need to convert this English phrase into mathematical notation

2. TRANSLATE the key phrase

• Focus on "84 less than \(\mathrm{x}\)"
• "Less than" means subtraction FROM the number mentioned
• "84 less than \(\mathrm{x}\)" means: start with \(\mathrm{x}\), then subtract 84
• This gives us: \(\mathrm{x - 84}\)

3. Set up the equation

• Since \(\mathrm{y}\) equals this expression: \(\mathrm{y = x - 84}\)
• This matches Choice D

Answer: D. \(\mathrm{y = x - 84}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse the word order and write \(\mathrm{y = 84 - x}\) instead of \(\mathrm{y = x - 84}\).

They might think "84 less than \(\mathrm{x}\)" means "84 minus \(\mathrm{x}\)" rather than "\(\mathrm{x}\) minus 84." This backwards interpretation comes from reading the phrase left-to-right instead of understanding that "less than" refers to subtraction FROM the second number mentioned.

This may lead them to calculate an expression that isn't among the choices, causing confusion and guessing.

Second Most Common Error:

Poor TRANSLATE reasoning: Students confuse "less than" with "more than" and write \(\mathrm{y = x + 84}\).

They might misread or misremember the problem statement, thinking \(\mathrm{y}\) is 84 MORE than \(\mathrm{x}\) instead of 84 LESS than \(\mathrm{x}\). This fundamental misunderstanding of the relationship direction is a classic reading comprehension error in math problems.

This may lead them to select Choice A (\(\mathrm{y = x + 84}\)).

The Bottom Line:

Success on this problem depends entirely on careful translation of mathematical language. The phrase "A is B less than C" always means \(\mathrm{A = C - B}\), not \(\mathrm{A = B - C}\).