\(\mathrm{q = r(5s + 7)}\)The given equation relates the distinct positive numbers q, r, and s. Which equation correctly expresses...

1. TRANSLATE the problem information

• Given equation: \(\mathrm{q = r(5s + 7)}\)
• Find: An expression for \(\mathrm{5s + 7}\) in terms of q and r
• What this tells us: We need to isolate the expression \(\mathrm{(5s + 7)}\) on one side

2. INFER the algebraic strategy

• Since \(\mathrm{q = r(5s + 7)}\), the expression \(\mathrm{(5s + 7)}\) is currently multiplied by r
• To isolate \(\mathrm{(5s + 7)}\), we need to "undo" this multiplication by dividing both sides by r
• This is valid because r is positive (given in the problem), so \(\mathrm{r ≠ 0}\)

3. SIMPLIFY by performing the division

• Start with: \(\mathrm{q = r(5s + 7)}\)
• Divide both sides by r: \(\frac{\mathrm{q}}{\mathrm{r}} = \frac{\mathrm{r(5s + 7)}}{\mathrm{r}}\)
• The r terms cancel on the right side: \(\frac{\mathrm{q}}{\mathrm{r}} = \mathrm{5s + 7}\)
• Rewrite: \(\mathrm{5s + 7} = \frac{\mathrm{q}}{\mathrm{r}}\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may think they need to multiply both sides by r instead of divide, possibly because they see r as a factor they need to "bring over" to the other side.

This incorrect reasoning leads them to: \(\mathrm{q·r = r^2(5s + 7)}\), which doesn't help isolate \(\mathrm{(5s + 7)}\). They might then try other manipulations and end up selecting Choice B (qr).

Second Most Common Error:

Poor INFER reasoning: Students correctly recognize they need to divide, but divide by the wrong variable (q instead of r).

Starting with \(\mathrm{q = r(5s + 7)}\), they might divide both sides by q to get: \(\mathrm{1 = \frac{r(5s + 7)}{q}}\), then rearrange to get \(\mathrm{5s + 7 = \frac{1}{r} = \frac{r}{q}}\). This leads them to select Choice D (r/q).

The Bottom Line:

This problem tests whether students can systematically think through inverse operations. The key insight is recognizing that when a desired expression is multiplied by something, you divide by that same thing to isolate it.