One pound of grapes costs $2. At this rate, how many dollars will c pounds of grapes cost?

1. TRANSLATE the problem information

• Given information:
  • 1 pound of grapes costs \(\$2\)
  • Need to find cost of c pounds

• What we're looking for: A mathematical expression for the cost of c pounds

2. INFER the relationship pattern

• This is a unit rate problem - we have a cost per unit and need total cost
• Pattern recognition:
  • If 1 pound costs \(\$2\), then 2 pounds cost \(2 \times \$2\)
  • If 1 pound costs \(\$2\), then 3 pounds cost \(3 \times \$2\)
  • If 1 pound costs \(\$2\), then c pounds cost \(\mathrm{c} \times \$2\)

3. TRANSLATE the pattern into mathematical notation

• c pounds cost \(\mathrm{c} \times \$2 = \$2\mathrm{c}\)

Answer: A. 2c

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students incorrectly think about combining the given information rather than recognizing the multiplication relationship in unit rates.

Instead of seeing "c pounds cost c times \(\$2\) each," they think "I have \(\$2\) and c pounds, so I'll add them: \(2 + \mathrm{c}\)"

This leads them to incorrectly choose option D.

Second Most Common Error:

Weak TRANSLATE skill: Students recognize division might be involved but apply it backwards.

They think "I need to divide the cost by the pounds" and write \(\frac{2}{\mathrm{c}}\) or \(\frac{\$2}{\mathrm{c}}\)

This represents cost per pound when c pounds cost \(\$2\) total, which is the opposite of what's given.

The Bottom Line:

The key is recognizing this as a unit rate multiplication: Total Cost = (Cost per pound) × (Number of pounds) = \(\$2\mathrm{c}\)