3a + 4b = 25 A shipping company charged a customer $25 to ship some small boxes and some large...

1. TRANSLATE the problem information

• Given information:
  • Equation: \(\mathrm{3a + 4b = 25}\)
  • \(\mathrm{a}\) = number of small boxes = 3
  • \(\mathrm{b}\) = number of large boxes = unknown
• We need to find the value of \(\mathrm{b}\)

2. INFER the solution approach

• Since we know the value of \(\mathrm{a}\), we can substitute it into the equation
• This will give us a simple equation with only one variable (\(\mathrm{b}\)) to solve

3. SIMPLIFY by substituting and solving

• Substitute \(\mathrm{a = 3}\) into the equation:

\(\mathrm{3(3) + 4b = 25}\)

• Multiply:

\(\mathrm{9 + 4b = 25}\)

• Subtract 9 from both sides:

\(\mathrm{4b = 16}\)

• Divide both sides by 4:

\(\mathrm{b = 4}\)

Answer: B. 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may confuse what the variables represent or misunderstand what they're solving for. Some students might try to solve for \(\mathrm{a}\) instead of \(\mathrm{b}\), or not realize they need to substitute the given value of \(\mathrm{a = 3}\).

This leads to confusion and random answer selection.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors during the substitution or algebraic manipulation steps. Common mistakes include:

• Calculating \(\mathrm{3(3)}\) as something other than 9
• Making errors when subtracting 9 from 25
• Dividing incorrectly: \(\mathrm{16 ÷ 4}\)

This may lead them to select Choice A (3), Choice C (5), or Choice D (6) depending on the specific calculation error.

The Bottom Line:

This problem tests whether students can correctly interpret a linear equation in context and execute basic substitution. The key insight is recognizing that when one variable's value is given, you simply substitute and solve for the other variable.