Keenan made 32 cups of vegetable broth. Keenan then filled x small jars and y large jars with all the...

1. TRANSLATE the equation components

• Given information:
  • Total broth: \(32\) cups
  • Equation: \(3\mathrm{x} + 5\mathrm{y} = 32\)
  • \(\mathrm{x}\) = number of small jars
  • \(\mathrm{y}\) = number of large jars

• What this tells us: The equation shows how \(32\) cups are distributed among the jars

2. INFER what the coefficients represent

• Since we're distributing \(32\) cups total among jars, the coefficients must represent how much each jar holds:
  • \(3\) = cups per small jar
  • \(5\) = cups per large jar

• This makes logical sense: different jar sizes hold different amounts

3. TRANSLATE what 5y means

• Now we can interpret \(5\mathrm{y}\):
  • \(5\) = cups per large jar
  • \(\mathrm{y}\) = number of large jars
  • \(5\mathrm{y}\) = (cups per jar) × (number of jars) = total cups in large jars

Answer: C. The total number of cups of vegetable broth in the large jars

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students focus only on the variable \(\mathrm{y}\) and ignore the coefficient \(5\)

They see \(\mathrm{y}\) represents "large jars" and immediately think \(5\mathrm{y}\) must also just mean "large jars" in some way. They miss that the coefficient \(5\) has its own meaning (capacity per jar) and that multiplying creates a new quantity (total capacity).

This may lead them to select Choice A (The number of large jars Keenan filled)

Second Most Common Error:

Poor INFER reasoning: Students don't recognize that coefficients represent jar capacities

Without understanding that \(3\) and \(5\) represent how much each jar holds, they can't properly interpret what \(3\mathrm{x}\) and \(5\mathrm{y}\) represent. They might think the coefficients are arbitrary or represent something else entirely.

This leads to confusion and guessing among the remaining answer choices.

The Bottom Line:

Success requires recognizing that in context problems, coefficients often represent rates, capacities, or unit quantities—not just mathematical placeholders. The key insight is that \(5\mathrm{y}\) combines the unit quantity (\(5\) cups per jar) with the count (\(\mathrm{y}\) jars) to give a total quantity.