A school allocates its $50,000 annual departmental budget between mathematics m and science s departments. School policy requires that mathematics...

1. TRANSLATE the key phrase

• The phrase "mathematics funding must exceed science funding by at least $800" needs to be converted to math notation
• "Exceed by at least" means the difference should be greater than or equal to
• Mathematics funding (m) minus science funding (s) must be ≥ $800

2. INFER what the inequality should look like

• We're looking for a difference, not a sum
• The difference \(\mathrm{m - s}\) should be at least $800
• "At least" means ≥, not ≤
• This gives us: \(\mathrm{m - s \geq 800}\)

3. TRANSLATE to eliminate wrong choices

• Choices C and D involve \(\mathrm{m + s}\) (total funding) - but we need difference, not sum
• Choice B uses ≤, which would mean math funding exceeds science by "at most" $800 - opposite of what we want

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "exceed by at least $800" as meaning the total budget should be at least $800, rather than focusing on the difference between the two amounts.

They might think: "The school needs at least $800 total for both departments," leading them to select Choice C (\(\mathrm{m + s \geq 800}\)) or get confused about whether it should be ≥ or ≤ for the total.

Second Most Common Error:

Poor TRANSLATE reasoning: Students correctly identify that we need a difference (\(\mathrm{m - s}\)) but get confused about the direction of the inequality, thinking "exceed by at least" means "exceed by no more than."

This backwards thinking leads them to select Choice B (\(\mathrm{m - s \leq 800}\)).

The Bottom Line:

The key challenge is accurately translating "exceed by at least" - students must recognize this means finding a difference (not a sum) and that the difference must be greater than or equal to (not less than or equal to) the specified amount.