A contest prize of $200 is divided between two winners. The first winner receives $x and the second winner receives...

1. TRANSLATE the problem information

• Given information:
  • Total prize = \(\$200\) (split between two winners)
  • First winner receives \(\$x\), second winner receives \(\$y\)
  • First winner receives \(\$20\) more than 3 times what second winner receives

• What this tells us mathematically:
  • \(\mathrm{x + y = 200}\)
  • \(\mathrm{x = 3y + 20}\)

2. INFER the solution approach

• We have two equations with two unknowns - this is a system of equations
• Since the second equation already isolates x, substitution method will work efficiently
• Strategy: substitute the expression for x into the first equation

3. SIMPLIFY using substitution

• Substitute \(\mathrm{x = 3y + 20}\) into \(\mathrm{x + y = 200}\):
  • \(\mathrm{(3y + 20) + y = 200}\)
  • \(\mathrm{4y + 20 = 200}\)
  • \(\mathrm{4y = 180}\)
  • \(\mathrm{y = 45}\)

4. SIMPLIFY to find x

• Now that \(\mathrm{y = 45}\), substitute back:
  • \(\mathrm{x = 3(45) + 20}\)
  • \(\mathrm{x = 135 + 20}\)
  • \(\mathrm{x = 155}\)

Answer: D. 155

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students struggle to convert "20 more than 3 times what the second winner receives" into the equation \(\mathrm{x = 3y + 20}\). They might write incorrect relationships like:

• \(\mathrm{x = 3y - 20}\) (subtracting instead of adding)
• \(\mathrm{x = 3(y + 20)}\) (misplacing the parentheses)
• \(\mathrm{3x + 20 = y}\) (completely backwards relationship)

Any of these translation errors leads to a different system of equations and an incorrect final answer that doesn't match any given choice, causing confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the equations but make algebraic errors when solving \(\mathrm{4y + 20 = 200}\). Common mistakes include:

• Forgetting to subtract 20 from both sides: \(\mathrm{4y = 200}\), so \(\mathrm{y = 50}\)
• Arithmetic errors: \(\mathrm{200 - 20 = 160}\) (instead of 180), leading to \(\mathrm{y = 40}\)

These errors cascade to wrong values of x, potentially leading them to select Choice B (140) or other incorrect options.

The Bottom Line:

This problem tests whether students can accurately translate complex verbal relationships into mathematical equations. The phrase "20 more than 3 times" requires careful parsing - students must recognize the order of operations in language and convert it precisely to algebra.