Question:In a contest, a participant's score S is calculated by S = 4c - w, where c is the number...

1. TRANSLATE the problem information

• Given information:
  • Scoring formula: \(\mathrm{S = 4c - w}\) (where \(\mathrm{c}\) = correct answers, \(\mathrm{w}\) = wrong answers)
  • Passing condition: \(\mathrm{S \gt 199}\)
  • Need to find which table has ALL pairs resulting in passing scores
• What this tells us: We need to calculate the score for every single \(\mathrm{(c, w)}\) pair in each table and check if all of them exceed 199.

2. SIMPLIFY by systematic calculation

• Strategy: Go through each table one by one, calculate \(\mathrm{S = 4c - w}\) for every pair, and immediately check if \(\mathrm{S \gt 199}\).

3. Check Choice A

• (50, 0): \(\mathrm{S = 4(50) - 0 = 200}\) ✓ (passes since \(\mathrm{200 \gt 199}\))
• (51, 8): \(\mathrm{S = 4(51) - 8 = 204 - 8 = 196}\) ✗ (fails since \(\mathrm{196 \lt 199}\))
• (52, 4): \(\mathrm{S = 4(52) - 4 = 208 - 4 = 204}\) ✓ (passes since \(\mathrm{204 \gt 199}\))

Since not all pairs pass, Choice A is eliminated.

4. Check Choice B

• (50, 1): \(\mathrm{S = 4(50) - 1 = 200 - 1 = 199}\) ✗ (fails since 199 is not \(\mathrm{\gt 199}\))
• (51, 4): \(\mathrm{S = 4(51) - 4 = 204 - 4 = 200}\) ✓ (passes since \(\mathrm{200 \gt 199}\))
• (52, 8): \(\mathrm{S = 4(52) - 8 = 208 - 8 = 200}\) ✓ (passes since \(\mathrm{200 \gt 199}\))

Since not all pairs pass, Choice B is eliminated.

5. Check Choice C

• (52, 0): \(\mathrm{S = 4(52) - 0 = 208}\) ✓ (passes since \(\mathrm{208 \gt 199}\))
• (50, 4): \(\mathrm{S = 4(50) - 4 = 200 - 4 = 196}\) ✗ (fails since \(\mathrm{196 \lt 199}\))
• (51, 8): \(\mathrm{S = 4(51) - 8 = 204 - 8 = 196}\) ✗ (fails since \(\mathrm{196 \lt 199}\))

Since not all pairs pass, Choice C is eliminated.

6. Check Choice D

• (50, 0): \(\mathrm{S = 4(50) - 0 = 200}\) ✓ (passes since \(\mathrm{200 \gt 199}\))
• (51, 4): \(\mathrm{S = 4(51) - 4 = 204 - 4 = 200}\) ✓ (passes since \(\mathrm{200 \gt 199}\))
• (52, 8): \(\mathrm{S = 4(52) - 8 = 208 - 8 = 200}\) ✓ (passes since \(\mathrm{200 \gt 199}\))

7. CONSIDER ALL CASES to verify

All three pairs in Choice D result in scores that exceed 199, making it the only table where all pairs result in passing scores.

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES skill: Students check only one or two pairs per table instead of systematically checking every single pair.

For example, they might see that (50, 0) in Choice A gives \(\mathrm{S = 200 \gt 199}\) and immediately assume Choice A is correct without checking the other two pairs. This leads them to miss that (51, 8) gives \(\mathrm{S = 196 \lt 199}\), which fails the requirement.

This may lead them to select Choice A (incorrect) or causes confusion when their 'first impression' doesn't match other calculations.

Second Most Common Error:

Misunderstanding inequality notation: Students think that \(\mathrm{S = 199}\) counts as 'passing' because they confuse \(\mathrm{S \gt 199}\) with \(\mathrm{S \geq 199}\).

When they see pairs like (50, 1) in Choice B giving \(\mathrm{S = 199}\), they mark it as passing instead of failing. This conceptual confusion about 'greater than' vs 'greater than or equal to' leads them to incorrect conclusions.

This may lead them to select Choice B (incorrect).

The Bottom Line:

This problem requires methodical checking of every single data point rather than spot-checking, combined with precise understanding of inequality conditions. Students who rush or make assumptions about 'close enough' will miss the correct answer.