A gardener buys two kinds of fertilizer. Fertilizer A contains 60% filler materials by weight and Fertilizer B contains 40%...

1. TRANSLATE the problem information

• Given information:
  • Fertilizer A: \(60\%\) filler materials by weight
  • Fertilizer B: \(40\%\) filler materials by weight
  • Total filler materials: \(240\) pounds
  • \(\mathrm{x}\) = pounds of Fertilizer A purchased
  • \(\mathrm{y}\) = pounds of Fertilizer B purchased

• What this tells us: We need to find how much filler comes from each fertilizer type

2. TRANSLATE percentages to filler amounts

• From x pounds of Fertilizer A:
  • Filler amount = \(60\% \times \mathrm{x} = 0.6\mathrm{x}\) pounds

• From y pounds of Fertilizer B:
  • Filler amount = \(40\% \times \mathrm{y} = 0.4\mathrm{y}\) pounds

3. INFER the relationship for total filler

• Since both fertilizers contribute to the 240 pounds of total filler:
  • Filler from A + Filler from B = Total filler
  • \(0.6\mathrm{x} + 0.4\mathrm{y} = 240\)

Answer: B. \(0.6\mathrm{x} + 0.4\mathrm{y} = 240\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students mix up which fertilizer has which percentage of filler materials.

They might think: "Fertilizer A has 60% filler, so that should be 0.6, but let me put that with y instead of x." This leads to the equation \(0.4\mathrm{x} + 0.6\mathrm{y} = 240\), where they've swapped the coefficients.

This may lead them to select Choice A (\(0.4\mathrm{x} + 0.6\mathrm{y} = 240\)).

Second Most Common Error:

Poor percentage-to-decimal conversion: Students use the percentage values directly instead of converting to decimals.

They think: "60% means 60, not 0.6" and write \(60\mathrm{x} + 40\mathrm{y} = 240\) or get confused about which fertilizer is which and write \(40\mathrm{x} + 60\mathrm{y} = 240\).

This may lead them to select Choice C (\(40\mathrm{x} + 60\mathrm{y} = 240\)) or Choice D (\(60\mathrm{x} + 40\mathrm{y} = 240\)).

The Bottom Line:

Success on this problem requires careful attention to detail when translating between English and math notation. Students must both convert percentages correctly AND keep track of which percentage belongs with which variable.