A group of 202 people went on an overnight camping trip, taking 60 tents with them. Some of the tents...

1. TRANSLATE the problem information

• Given information:
  • 202 total people
  • 60 total tents
  • Some tents hold 2 people, others hold 4 people
  • All tents filled to capacity, everyone gets a tent
  • Find: number of 2-person tents
• What we need: Two variables for the two types of tents

2. INFER the mathematical approach

• This is a system of equations problem with two unknowns
• Let \(\mathrm{x}\) = number of 2-person tents, \(\mathrm{y}\) = number of 4-person tents
• We need equations for: total tents AND total people

3. TRANSLATE constraints into equations

Set up the system:

• Total tents equation: \(\mathrm{x + y = 60}\)
• Total people equation: \(\mathrm{2x + 4y = 202}\)

4. INFER the solution strategy

• Use substitution method since the first equation easily gives us y in terms of x
• From \(\mathrm{x + y = 60}\), we get: \(\mathrm{y = 60 - x}\)

5. SIMPLIFY using substitution

Substitute \(\mathrm{y = 60 - x}\) into the people equation:

\(\mathrm{2x + 4(60 - x) = 202}\)

\(\mathrm{2x + 240 - 4x = 202}\)

\(\mathrm{-2x + 240 = 202}\)

\(\mathrm{-2x = -38}\)

\(\mathrm{x = 19}\)

6. Verify the solution

If \(\mathrm{x = 19}\), then \(\mathrm{y = 60 - 19 = 41}\)

Check: \(\mathrm{2(19) + 4(41) = 38 + 164 = 202}\) ✓

Answer: C. 19

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students struggle to set up the correct system of equations, often missing that they need BOTH a tent equation and a people equation. Some students only set up one equation like \(\mathrm{2x + 4y = 202}\) and then get stuck not knowing how to solve for two variables.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the system but make algebraic errors during substitution. For example, they might incorrectly distribute: \(\mathrm{2x + 4(60 - x) = 202}\) becomes \(\mathrm{2x + 240 + 4x = 202}\) (wrong sign), leading to \(\mathrm{6x = -38}\), so \(\mathrm{x \approx -6.3}\), which makes no sense.

This causes them to doubt their approach and potentially guess, or they might work backwards from answer choices and accidentally select Choice B (20) through calculation errors.

The Bottom Line:

This problem tests whether students can recognize a system of equations scenario in a real-world context and execute the algebraic solution correctly. The camping context makes the mathematical relationships less obvious than pure algebraic problems.