The given equation c + 38 = t/r relates the positive numbers c, t, and r. Which equation correctly expresses...

1. TRANSLATE the problem information

• Given equation: \(\mathrm{c + 38 = \frac{t}{r}}\)
• Goal: Express \(\mathrm{t}\) in terms of \(\mathrm{c}\) and \(\mathrm{r}\)

2. INFER the solving strategy

• The variable \(\mathrm{t}\) appears in a fraction on the right side
• To isolate \(\mathrm{t}\), I need to eliminate the fraction by multiplying both sides by \(\mathrm{r}\)
• This will "undo" the division by \(\mathrm{r}\)

3. SIMPLIFY by multiplying both sides by r

• Starting equation: \(\mathrm{c + 38 = \frac{t}{r}}\)
• Multiply both sides by \(\mathrm{r}\): \(\mathrm{r(c + 38) = r \cdot \frac{t}{r}}\)
• Right side simplifies: \(\mathrm{r \cdot \frac{t}{r} = t}\)
• So now I have: \(\mathrm{r(c + 38) = t}\)

4. SIMPLIFY by applying the distributive property

• Current equation: \(\mathrm{t = r(c + 38)}\)
• Apply distributive property: \(\mathrm{t = r \cdot c + r \cdot 38}\)
• Simplify: \(\mathrm{t = cr + 38r}\)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students correctly multiply both sides by \(\mathrm{r}\) to get \(\mathrm{t = r(c + 38)}\), but then fail to fully distribute the \(\mathrm{r}\) to both terms inside the parentheses.

Instead of getting \(\mathrm{t = cr + 38r}\), they might write \(\mathrm{t = cr + 38}\), forgetting that the \(\mathrm{r}\) must multiply the entire expression \(\mathrm{(c + 38)}\). This incomplete distribution leads them to select Choice B \(\mathrm{(cr + 38)}\).

Second Most Common Error:

Poor INFER reasoning: Students attempt to solve for \(\mathrm{t}\) by working with the fraction directly, perhaps trying to "flip" the equation or manipulate it in ways that don't lead to a clear solution path.

This confusion about strategy causes them to get stuck and randomly select an answer, or they might incorrectly think the answer should keep the fraction form and select Choice A \(\mathrm{(\frac{c + 38}{r})}\).

The Bottom Line:

This problem tests whether students can systematically eliminate fractions from equations and properly apply the distributive property - two fundamental algebraic skills that require careful, step-by-step execution.