For a camping trip a group bought x one-liter bottles of water and y three-liter bottles of water, for a...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{x}\) = number of one-liter bottles
  • \(\mathrm{y}\) = number of three-liter bottles
  • Total water = \(240\) liters

2. INFER how to calculate total volume

• Key insight: Total volume = (number of bottles) × (capacity per bottle) for each type
• We need to find the volume contribution from each bottle type separately

3. TRANSLATE each bottle type's contribution

• From \(\mathrm{x \times 1 = x}\) one-liter bottles: \(\mathrm{x}\) liters
• From \(\mathrm{y \times 3 = 3y}\) three-liter bottles: \(\mathrm{3y}\) liters

4. Set up the equation

• Total volume = volume from one-liter bottles + volume from three-liter bottles
• \(\mathrm{240 = x + 3y}\)
• Therefore: \(\mathrm{x + 3y = 240}\)

Answer: A. \(\mathrm{x + 3y = 240}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students translate "\(\mathrm{x}\) one-liter bottles and \(\mathrm{y}\) three-liter bottles" as simply "\(\mathrm{x + y}\)" without considering the different capacities.

They think: "\(\mathrm{x}\) bottles plus \(\mathrm{y}\) bottles equals some total, so the equation should be \(\mathrm{x + y = 240}\)."

This leads them to select Choice B (\(\mathrm{x + y = 240}\)).

Second Most Common Error:

Poor TRANSLATE reasoning: Students mix up which variable gets which coefficient, incorrectly thinking that since there are 3-liter bottles, the \(\mathrm{x}\) should get multiplied by 3.

They might reason: "There are bigger bottles involved, so I need a 3 somewhere with \(\mathrm{x}\)."

This leads them to select Choice D (\(\mathrm{3x + y = 240}\)).

The Bottom Line:

This problem requires careful attention to units and the distinction between counting bottles versus measuring total volume. Students must recognize that the variable represents the count, but the equation must account for each bottle's individual capacity.