In 2008, Zinah earned 14% more than in 2007, and in 2009 Zinah earned 4% more than in 2008. If...

1. TRANSLATE the problem information

• Given information:
  • 2008 earnings: 14% more than 2007
  • 2009 earnings: 4% more than 2008
  • Need to find: \(\mathrm{y}\) where \(\mathrm{2009 = y \times 2007}\)

• What this tells us: We need to track how earnings change through each year, then compare 2009 to the original 2007 amount.

2. TRANSLATE percentage increases into multipliers

• 14% increase means \(\mathrm{new\ amount = 1.14 \times original\ amount}\)
• 4% increase means \(\mathrm{new\ amount = 1.04 \times original\ amount}\)

3. Set up variables and equations

• Let \(\mathrm{A}\) = Zinah's 2007 earnings
• 2008 earnings = \(\mathrm{1.14A}\) (14% more than 2007)
• 2009 earnings = \(\mathrm{1.04 \times (2008\ earnings) = 1.04 \times 1.14A}\)

4. SIMPLIFY to find the relationship

\(\mathrm{2009\ earnings = 1.04 \times 1.14A}\)

Calculate: \(\mathrm{1.04 \times 1.14 = 1.1856}\) (use calculator)

So \(\mathrm{2009\ earnings = 1.1856A}\)

5. Find y

Since \(\mathrm{2009 = y \times 2007}\): \(\mathrm{1.1856A = y \times A}\)

Therefore: \(\mathrm{y = 1.1856}\)

Answer: D. 1.1856

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students incorrectly think they can simply add the percentages together: \(\mathrm{14\% + 4\% = 18\%}\), so \(\mathrm{y = 1.18}\).

This reasoning misses that the 4% increase applies to the already-increased 2008 amount, not the original 2007 amount. The increases compound rather than simply add.

This leads them to select Choice C (1.1800).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the problem as \(\mathrm{1.14 \times 1.04}\) but make calculation errors with the decimal multiplication.

Common mistakes include getting 1.0056 (possibly from \(\mathrm{1.14 \times 0.04}\)) or other decimal errors that don't result in the correct product.

This may lead them to select Choice B (1.0056) or causes confusion and guessing.

The Bottom Line:

This problem requires understanding that percentage increases compound (multiply) rather than simply add together, combined with careful decimal arithmetic to avoid calculation errors.