Let k be a positive number such that sqrt(5k) = 45/(sqrt(5k)). What is the value of 15k?...

1. TRANSLATE the problem information

• Given equation: \(\sqrt{5\mathrm{k}} = \frac{45}{\sqrt{5\mathrm{k}}}\)
• Find: The value of \(15\mathrm{k}\)

2. INFER the solution strategy

• The equation has a square root in both the numerator (left side) and denominator (right side)
• Key insight: Multiply both sides by \(\sqrt{5\mathrm{k}}\) to eliminate the fraction and create a solvable equation
• This will give us \((\sqrt{5\mathrm{k}})^2\) on the left and simplify the right side

3. SIMPLIFY by multiplying both sides by \(\sqrt{5\mathrm{k}}\)

• Left side: \(\sqrt{5\mathrm{k}} \times \sqrt{5\mathrm{k}} = (\sqrt{5\mathrm{k}})^2 = 5\mathrm{k}\)
• Right side: \(\frac{45}{\sqrt{5\mathrm{k}}} \times \sqrt{5\mathrm{k}} = 45 \times \frac{\sqrt{5\mathrm{k}}}{\sqrt{5\mathrm{k}}} = 45 \times 1 = 45\)
• Result: \(5\mathrm{k} = 45\)

4. INFER the direct path to the answer

• The problem asks for \(15\mathrm{k}\), not k
• Notice that \(15\mathrm{k} = 3 \times 5\mathrm{k}\)
• Since \(5\mathrm{k} = 45\), then \(15\mathrm{k} = 3 \times 45 = 135\)

Answer: C. 135

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the efficient strategy of multiplying both sides by \(\sqrt{5\mathrm{k}}\). Instead, they might attempt to manipulate the equation in more complex ways, such as trying to isolate k by squaring both sides incorrectly or getting confused by the fraction with a radical denominator. This leads to algebraic complications and often incorrect equations that don't lead to the right answer choices.

Second Most Common Error:

Inefficient INFER reasoning: Students correctly solve the equation to get \(5\mathrm{k} = 45\), but then unnecessarily solve for \(\mathrm{k} = 9\) and calculate \(15\mathrm{k} = 15 \times 9 = 135\). While this gets the correct answer, it represents extra work and creates more opportunities for arithmetic errors. Some students might make mistakes in this final multiplication step.

The Bottom Line:

The key challenge is recognizing that multiplying both sides by \(\sqrt{5\mathrm{k}}\) creates a clean, solvable equation, and then realizing you can work directly with the \(5\mathrm{k}\) result rather than solving for k first. Students who miss this strategic insight often get bogged down in unnecessary algebraic complexity.