A researcher surveyed people in two districts about support for a new policy. In the first district, 40% of those...

1. TRANSLATE the problem information

• Given information:
  • First district: \(40\% \text{ of } \mathrm{a}\) people approve the policy
  • Second district: \(80\% \text{ of } \mathrm{b}\) people approve the policy
  • Need at least 600 total approvals from both districts

2. TRANSLATE each district's approvals

• From first district: \(40\% \text{ of } \mathrm{a} = 0.4\mathrm{a}\) people approve
• From second district: \(80\% \text{ of } \mathrm{b} = 0.8\mathrm{b}\) people approve
• Total approvals from both districts = \(0.4\mathrm{a} + 0.8\mathrm{b}\)

3. INFER the inequality constraint

• "At least 600" means the total must be greater than or equal to 600
• This gives us: \(0.4\mathrm{a} + 0.8\mathrm{b} \geq 600\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students mix up the approval percentages between districts, thinking the first district has 80% approval and the second has 40%.

This leads them to write \(0.8\mathrm{a} + 0.4\mathrm{b} \geq 600\), causing them to select Choice B (\(0.8\mathrm{a} + 0.4\mathrm{b} \leq 600\)) if they also make the inequality direction error, or get confused about which choice matches their work.

Second Most Common Error:

Poor TRANSLATE reasoning: Students convert 40% and 80% incorrectly, treating them as whole numbers (40 and 80) rather than decimals (0.4 and 0.8).

This makes them write \(40\mathrm{a} + 80\mathrm{b} \geq 600\), leading them to select Choice D (\(40\mathrm{a} + 80\mathrm{b} \geq 600\)).

The Bottom Line:

This problem tests your ability to systematically translate word descriptions into mathematical notation. The key is carefully tracking which percentage goes with which district and properly converting percentages to decimals for calculation.