A courier's weekly earnings are based on the number of small envelopes and large packages she delivers. She delivered x...

1. TRANSLATE the equation structure

• Given information:
  • Equation: \(\mathrm{2.75x + 7y = 217}\)
  • \(\mathrm{x}\) = number of small envelopes delivered
  • \(\mathrm{y}\) = number of large packages delivered
  • 217 = total weekly earnings in dollars

2. INFER what each term must represent

• Since this is a total earnings equation, it follows the pattern:
  (earnings from item 1) + (earnings from item 2) = total earnings
• Each earnings term has the structure: \(\mathrm{(rate\:per\:item) \times (quantity) = total\:from\:that\:item}\)

3. TRANSLATE each specific term

• Term \(\mathrm{2.75x}\): This must be $2.75 per small envelope × \(\mathrm{x}\) envelopes = total earnings from small envelopes
• Term \(\mathrm{7y}\): This must be $7 per large package × \(\mathrm{y}\) packages = total earnings from large packages
• The coefficient \(\mathrm{7}\) tells us she earns $7 per large package
• The full term \(\mathrm{7y}\) tells us the total dollars earned from all large packages

Answer: B - The total amount, in dollars, earned from delivering large packages

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students focus only on the coefficient \(\mathrm{7}\) and think it represents "the amount earned for each large package delivered."

They read \(\mathrm{7y}\) and only see the "7" part, forgetting that the complete term \(\mathrm{7y}\) represents a total amount (rate × quantity). This leads them to select Choice A ($7 per package) instead of recognizing that \(\mathrm{7y}\) represents the total earnings from all packages.

Second Most Common Error:

Poor INFER reasoning about equation structure: Students don't recognize that in earning equations, individual terms represent subtotals that add up to the grand total.

They might confuse which variable represents which type of delivery, or misunderstand that \(\mathrm{7y}\) involves both the rate AND the quantity. This causes them to get stuck and guess among the remaining choices.

The Bottom Line:

This problem requires students to distinguish between a coefficient (unit rate) and a complete term (total amount). The key insight is recognizing that \(\mathrm{7y = (rate\:per\:package) \times (number\:of\:packages) = total\:earnings\:from\:packages)}\).