Each volunteer can stuff 220 envelopes in 1 hour, working at a constant rate. A campaign needs to stuff e...

1. TRANSLATE the problem information

• Given information:
  • Each volunteer stuffs 220 envelopes per hour
  • First mailing needs e envelopes
  • Second mailing needs e envelopes
  • All work must be completed in 1 hour
  • Find equation for number of volunteers v needed

• What this tells us:
  • Total envelopes needed: \(\mathrm{e + e = 2e}\)
  • Time available: 1 hour

2. INFER the approach

• Key insight: Total work capacity must equal total work needed
• In 1 hour, v volunteers will stuff \(\mathrm{220v}\) envelopes total
• This must equal the \(\mathrm{2e}\) envelopes that need stuffing
• Set up equation: \(\mathrm{220v = 2e}\)

3. SIMPLIFY to solve for v

• Starting from \(\mathrm{220v = 2e}\)
• Divide both sides by 220: \(\mathrm{v = \frac{2e}{220}}\)
• Simplify the fraction: \(\mathrm{v = \frac{e}{110}}\)

Answer: B. \(\mathrm{v = \frac{e}{110}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students miss that there are TWO mailings, each requiring e envelopes.

They read "e envelopes for a first mailing and the same number again for a second mailing" but only process the first part, thinking total envelopes = e instead of 2e. This leads them to set up \(\mathrm{220v = e}\), giving \(\mathrm{v = \frac{e}{220}}\).

This may lead them to select Choice A (\(\mathrm{v = \frac{e}{220}}\)).

Second Most Common Error:

Poor INFER reasoning: Students confuse the relationship between rate, workers, and total work.

They might think the equation should be "volunteers needed = rate/total work" instead of recognizing that "total capacity = total work needed." This backward thinking leads them to \(\mathrm{v = \frac{220}{e}}\) or similar incorrect setups.

This may lead them to select Choice C (\(\mathrm{v = \frac{220}{e}}\)).

The Bottom Line:

This problem requires careful reading to catch the "two mailings" detail and solid understanding of how work rates combine when multiple workers are involved. Students who rush through the reading or don't have a clear mental model of work rate relationships will struggle.