A company that provides whale-watching tours takes groups of 21 people at a time. The company's revenue is 80 dollars...

1. TRANSLATE the problem information

• Given information:
  • Total people per group: 21
  • Adult ticket price: \(\$80\)
  • Child ticket price: \(\$60\)
  • Total revenue for this group: \(\$1,440\)
  • Need to find: number of children

• What this tells us: We need two equations since we have two unknowns (adults and children)

2. TRANSLATE into mathematical equations

• Let \(\mathrm{x}\) = number of children, \(\mathrm{y}\) = number of adults
• People constraint: \(\mathrm{x + y = 21}\)
• Revenue constraint: \(\mathrm{60x + 80y = 1,440}\)

3. INFER the solution approach

• We have a system of linear equations with two unknowns
• Can use either substitution or elimination method
• Substitution looks simpler since the first equation easily gives us \(\mathrm{y}\) in terms of \(\mathrm{x}\)

4. SIMPLIFY using substitution method

• From \(\mathrm{x + y = 21}\), we get: \(\mathrm{y = 21 - x}\)
• Substitute into revenue equation:
  \(\mathrm{60x + 80(21 - x) = 1,440}\)
• Distribute:
  \(\mathrm{60x + 1,680 - 80x = 1,440}\)
• Combine like terms:
  \(\mathrm{-20x + 1,680 = 1,440}\)
• Solve:
  \(\mathrm{-20x = -240}\), so \(\mathrm{x = 12}\)

5. Verify the solution

• If \(\mathrm{x = 12}\) children, then \(\mathrm{y = 21 - 12 = 9}\) adults
• Revenue check: \(\mathrm{60(12) + 80(9) = 720 + 720 = 1,440}\) ✓

Answer: C. 12

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students struggle to set up the correct system of equations, especially the revenue equation. They might write something like "60 + 80 = 1,440" or confuse which variable represents what quantity.

This fundamental translation error makes systematic solution impossible, leading to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the equations correctly but make sign errors during algebraic manipulation, particularly when distributing \(\mathrm{80(21 - x)}\) or combining like terms with \(\mathrm{-20x}\).

A common mistake is getting \(\mathrm{20x = 240}\) instead of \(\mathrm{-20x = -240}\), leading them to select Choice B (9) - which is actually the number of adults, not children.

The Bottom Line:

This problem requires strong equation setup skills and careful algebraic manipulation. Students must clearly distinguish between the two unknowns and maintain accuracy through multi-step algebra to avoid selecting the "wrong variable" answer choice.