Question:z = m - mrThe given equation relates positive numbers z, m, and r, where r ≠ 1. Which of...

1. TRANSLATE the problem information

• Given equation: \(\mathrm{z = m - mr}\)
• Need to express m in terms of z and r
• Constraint: \(\mathrm{r ≠ 1}\) (ensures our final answer will be valid)

2. INFER the solution strategy

• I notice both terms on the right side contain the variable m
• This suggests I should factor out m as a common factor
• Once factored, I can isolate m through division

3. SIMPLIFY by factoring out the common factor

Starting with: \(\mathrm{z = m - mr}\)

Factor out m from the right side: \(\mathrm{z = m(1 - r)}\)

4. SIMPLIFY by isolating the variable

Divide both sides by (1 - r):

\(\mathrm{\frac{z}{1 - r} = m}\)

Therefore: \(\mathrm{m = \frac{z}{1 - r}}\)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students don't recognize the factoring strategy and instead try to isolate m by moving terms around. They might add mr to both sides, getting \(\mathrm{z + mr = m}\), then incorrectly think this solves the problem.

This may lead them to select Choice A (\(\mathrm{m = z + mr}\)).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly factor to get \(\mathrm{z = m(1 - r)}\) but make a sign error when isolating m. They might flip the fraction incorrectly and write \(\mathrm{m = \frac{z}{r - 1}}\) instead of \(\mathrm{m = \frac{z}{1 - r}}\).

This may lead them to select Choice C (\(\mathrm{m = \frac{z}{r - 1}}\)).

The Bottom Line:

This problem tests whether students can recognize when factoring is the most efficient strategy and whether they can execute the algebraic steps without sign errors. The key insight is seeing that factoring out the common variable makes the isolation step straightforward.