If 54/z = 6z, what is the value of 15z^2?

1. INFER the strategic approach

• Given: \(\frac{54}{\mathrm{z}} = 6\mathrm{z}\)
• Target: Find \(15\mathrm{z}^2\)
• Key insight: We need \(\mathrm{z}^2\), not z, so we should solve for \(\mathrm{z}^2\) directly rather than finding z first

2. SIMPLIFY by eliminating the fraction

• Multiply both sides by z: \(\frac{54}{\mathrm{z}} \times \mathrm{z} = 6\mathrm{z} \times \mathrm{z}\)
• This gives us: \(54 = 6\mathrm{z}^2\)
• The fraction is now eliminated, making the equation easier to work with

3. SIMPLIFY to isolate z²

• Divide both sides by 6: \(54 \div 6 = 6\mathrm{z}^2 \div 6\)
• This gives us: \(\mathrm{z}^2 = 9\)

4. SIMPLIFY to find the final answer

• Substitute \(\mathrm{z}^2 = 9\) into \(15\mathrm{z}^2\):
• \(15\mathrm{z}^2 = 15(9) = 135\)

Answer: D. 135

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students try to solve for z first instead of recognizing they can work directly with \(\mathrm{z}^2\)

Students might take the square root: \(\mathrm{z}^2 = 9\), so \(\mathrm{z} = \pm3\), then calculate \(15\mathrm{z}^2 = 15(9) = 135\). While this still gives the correct answer, it's an unnecessary extra step that increases chances for sign errors. However, some students get confused about which value of z to use and may incorrectly calculate \(15\mathrm{z}^2\) using just one value of z rather than \(\mathrm{z}^2\).

This confusion may lead them to calculate incorrectly or guess among the choices.

Second Most Common Error:

Poor SIMPLIFY execution: Making arithmetic errors when multiplying both sides by z or when performing the final calculations

Students might incorrectly multiply \(\frac{54}{\mathrm{z}} \times \mathrm{z}\), thinking it gives 54z instead of 54, or make errors like \(54 \div 6 = 8\). These calculation mistakes lead to wrong values for \(\mathrm{z}^2\) and ultimately incorrect final answers.

This may lead them to select Choice A (9) if they think \(\mathrm{z}^2 = 9\) is the final answer, or Choice B (54) if they make certain arithmetic errors.

The Bottom Line:

This problem tests whether students can work strategically with equations involving both fractions and quadratic expressions, requiring them to see that solving for \(\mathrm{z}^2\) directly is more efficient than solving for z first.