A number w is at least 13 more than 3 times the value of z. If the value of z...

1. TRANSLATE the problem information

• Given information:
  • w is at least 13 more than 3 times the value of z
  • \(\mathrm{z = 4}\)
  • Need to find the smallest possible value of w

• What this tells us: We need to convert the verbal description into a mathematical inequality, then solve for w.

2. TRANSLATE the verbal expression into mathematical notation

• 'at least' means \(\geq\) (greater than or equal to)
• '13 more than 3 times the value of z' means \(\mathrm{3z + 13}\)
• Therefore: \(\mathrm{w \geq 3z + 13}\)

3. SIMPLIFY by substituting the known value

• Substitute \(\mathrm{z = 4}\) into our inequality:
  \(\mathrm{w \geq 3(4) + 13}\)
  \(\mathrm{w \geq 12 + 13}\)
  \(\mathrm{w \geq 25}\)

4. INFER the final answer

• Since w must be greater than or equal to 25, and we want the smallest possible value
• The smallest value that satisfies \(\mathrm{w \geq 25}\) is exactly 25

Answer: B) 25

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students struggle with converting 'at least 13 more than 3 times the value of z' into proper mathematical notation. They might write \(\mathrm{w = 3z + 13}\) (using equals instead of \(\geq\)) or incorrectly structure the expression as \(\mathrm{w \geq 13 + 3z}\), missing the multiplication priority.

This fundamental translation error leads to either getting stuck or proceeding with wrong mathematics, causing confusion and guessing.

Second Most Common Error:

Poor INFER reasoning: Students correctly get to \(\mathrm{w \geq 25}\) but then think they need to find a value larger than 25, not understanding that 'smallest possible value' with \(\geq 25\) is exactly 25. They might select Choice C (27) thinking 25 doesn't count because it's not 'more than' 25.

The Bottom Line:

This problem tests your ability to translate complex verbal mathematical relationships into inequalities. The key insight is recognizing that 'at least' creates a boundary condition, and the 'smallest possible value' that satisfies that boundary is the boundary value itself.