A clothing store is having a sale on shirts and pants. During the sale, the cost of each shirt is...

1. TRANSLATE the given information into mathematical expressions

• Given information:
  • Each shirt costs $15, Geoff buys s shirts → shirt costs = \(\mathrm{15s}\)
  • Each pair of pants costs $25, Geoff buys p pants → pants costs = \(\mathrm{25p}\)
  • Geoff can spend "at most $120" → total spending \(\leq\) 120

2. INFER how to combine the costs

• Total spending = shirt costs + pants costs
• Total spending = \(\mathrm{15s + 25p}\)

3. TRANSLATE the spending constraint into an inequality

• "At most $120" means the total cannot exceed $120
• This translates to: \(\mathrm{15s + 25p \leq 120}\)

Answer: A. \(\mathrm{15s + 25p \leq 120}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse "at most" with "at least"

They might think "at most $120" means Geoff must spend at least $120, leading them to write \(\mathrm{15s + 25p \geq 120}\).

This may lead them to select Choice B (\(\mathrm{15s + 25p \geq 120}\))

Second Most Common Error:

Poor TRANSLATE reasoning: Students mix up the prices of shirts and pants

They might reverse the coefficients, thinking shirts cost $25 and pants cost $15, creating the expression \(\mathrm{25s + 15p}\) instead of \(\mathrm{15s + 25p}\).

This may lead them to select Choice C (\(\mathrm{25s + 15p \leq 120}\)) or Choice D (\(\mathrm{25s + 15p \geq 120}\))

The Bottom Line:

This problem tests whether students can accurately translate verbal constraints into mathematical inequalities while keeping track of which price goes with which item. The key insight is recognizing that "at most" always means "\(\leq\)" (less than or equal to).