The equation 7g + 7b = 840 represents the number of blue tiles, b, and the number of green tiles,...

1. TRANSLATE the problem information

• Given information:
  • Equation: \(\mathrm{7g + 7b = 840}\)
  • The artist needs 71 blue tiles, which means \(\mathrm{b = 71}\)
  • Need to find: \(\mathrm{g}\) (number of green tiles)

2. INFER the solution approach

• Since we know the value of \(\mathrm{b}\) and have an equation with both \(\mathrm{g}\) and \(\mathrm{b}\), substitution is the most direct method
• We'll substitute \(\mathrm{b = 71}\) into the equation and solve for \(\mathrm{g}\)

3. SIMPLIFY through substitution and algebraic steps

• Substitute \(\mathrm{b = 71}\) into \(\mathrm{7g + 7b = 840}\):
  \(\mathrm{7g + 7(71) = 840}\)
• Calculate \(\mathrm{7 × 71 = 497}\):
  \(\mathrm{7g + 497 = 840}\)
• Subtract 497 from both sides:
  \(\mathrm{7g = 840 - 497}\)
  \(\mathrm{7g = 343}\)
• Divide both sides by 7:
  \(\mathrm{g = 343 ÷ 7 = 49}\)

Answer: 49

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students may not clearly identify what \(\mathrm{b = 71}\) means in the context of the equation, or they might confuse which variable represents which type of tile.

They might substitute incorrectly (perhaps using \(\mathrm{g = 71}\) instead of \(\mathrm{b = 71}\)) or fail to recognize that they have enough information to substitute. This leads to confusion and guessing.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students make arithmetic errors during the multi-step algebraic process, particularly when calculating \(\mathrm{7 × 71 = 497}\) or when performing \(\mathrm{343 ÷ 7}\).

Common arithmetic mistakes include getting \(\mathrm{7 × 71 = 497}\) wrong (maybe calculating it as 427 or 571) or making errors in the subtraction \(\mathrm{840 - 497}\). This leads them to arrive at incorrect values for \(\mathrm{g}\).

The Bottom Line:

This problem tests whether students can connect word problems to algebraic equations and execute substitution correctly. The key challenge is recognizing that having a specific value for one variable (\(\mathrm{b = 71}\)) allows you to solve for the other variable through direct substitution.