The equation (15p - 5q)/5 = 3sqrt(t) + 1 relates the distinct positive real numbers p, q, and t. Which...

1. SIMPLIFY the left side of the equation

• Given: \(\frac{15p - 5q}{5} = 3\sqrt{t} + 1\)
• Distribute the division: \(\frac{15p - 5q}{5} = \frac{15p}{5} - \frac{5q}{5} = 3p - q\)
• The equation becomes: \(3p - q = 3\sqrt{t} + 1\)

2. INFER the strategy to isolate the radical term

• Goal: Get \(\sqrt{t}\) by itself so we can square both sides
• First step: Move the constant term to isolate \(3\sqrt{t}\)

3. SIMPLIFY by moving the constant term

• Subtract 1 from both sides: \(3p - q - 1 = 3\sqrt{t}\)
• Now we have the radical term isolated on one side

4. SIMPLIFY to get √t alone

• Divide both sides by 3: \(\frac{3p - q - 1}{3} = \sqrt{t}\)
• Now \(\sqrt{t}\) is completely isolated

5. SIMPLIFY by eliminating the square root

• Square both sides: \(t = \left[\frac{3p - q - 1}{3}\right]^2\)
• This matches answer choice (B)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when simplifying \(\frac{15p - 5q}{5}\), often getting \(3p - q\) + something wrong or forgetting to distribute the division properly.

For example, they might write \(\frac{15p - 5q}{5} = 15p - q\) or \(3p - 5q\), leading to completely incorrect subsequent steps. This leads to confusion and guessing since none of their algebra will match the answer choices.

Second Most Common Error:

Poor INFER reasoning about equation solving strategy: Students attempt to square both sides too early, before properly isolating the radical term.

They might try to square \(3p - q = 3\sqrt{t} + 1\) immediately, creating a complex expression \((3\sqrt{t} + 1)^2 = 9t + 6\sqrt{t} + 1\) that's much harder to work with. This may lead them to select Choice A \((3p - q - 1)^2\) because they see a similar structure but haven't properly isolated the variable.

The Bottom Line:

This problem tests systematic algebraic manipulation skills. Success requires methodically isolating the radical term before attempting to eliminate it, combined with careful arithmetic throughout multiple steps.