The expression 0.35x represents the result of decreasing a positive quantity by what percent?

1. TRANSLATE the problem information

• Given: The expression \(0.35\mathrm{x}\) represents a positive quantity after being decreased by some percent
• Need to find: What percent decrease produces \(0.35\mathrm{x}\)?

2. INFER the mathematical setup

• When we decrease a quantity x by n percent, the result is: \(\mathrm{x}(1 - \frac{\mathrm{n}}{100})\)
• Since our result is \(0.35\mathrm{x}\), we need to solve: \(\mathrm{x}(1 - \frac{\mathrm{n}}{100}) = 0.35\mathrm{x}\)

3. SIMPLIFY to solve for the percent

• Divide both sides by x: \(1 - \frac{\mathrm{n}}{100} = 0.35\)
• Subtract 1 from both sides: \(-\frac{\mathrm{n}}{100} = 0.35 - 1 = -0.65\)
• Multiply by -100: \(\mathrm{n} = 65\)

4. Verify the answer

• Decreasing x by 65% gives us: \(\mathrm{x} - 0.65\mathrm{x} = 0.35\mathrm{x}\) ✓

Answer: D. 65%

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often confuse what \(0.35\mathrm{x}\) represents. They might think "\(0.35\mathrm{x}\) means we decreased by 35%" because they focus on the 0.35 coefficient without understanding that this represents what's LEFT after the decrease, not what was taken away.

This leads them to select Choice B (35%) by incorrectly reasoning that the decimal 0.35 directly corresponds to 35%.

Second Most Common Error:

Incomplete SIMPLIFY execution: Students correctly set up the equation but make algebraic errors. For example, from \(1 - \frac{\mathrm{n}}{100} = 0.35\), they might incorrectly solve to get \(\mathrm{n} = 3.5\) by confusing the steps or mishandling the decimal arithmetic.

This may lead them to select Choice A (3.5%) due to calculation mistakes.

The Bottom Line:

The key insight is recognizing that \(0.35\mathrm{x}\) represents what REMAINS after the decrease, not the amount that was decreased. Students must understand that if 35% remains, then 65% was removed.