\(\mathrm{Q = R(S + 12)}\) The given equation relates the positive numbers Q, R, and S. Which equation correctly expresses...

1. TRANSLATE the problem goal

• Given equation: \(\mathrm{Q = R(S + 12)}\)
• Goal: Express S in terms of Q and R (isolate S on one side)

2. SIMPLIFY using division property of equality

• Divide both sides by R: \(\mathrm{Q/R = S + 12}\)
• This removes R from the right side, leaving us closer to isolating S

3. SIMPLIFY using subtraction property of equality

• Subtract 12 from both sides: \(\mathrm{Q/R - 12 = S}\)
• This isolates S completely

4. Rewrite in standard form

• \(\mathrm{S = Q/R - 12}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make a sign error when dealing with the constant term.

After correctly getting \(\mathrm{Q/R = S + 12}\), they might add 12 to both sides instead of subtracting, thinking they need to "get rid of the 12." This gives them \(\mathrm{Q/R + 12 = S}\), leading them to select Choice A (\(\mathrm{S = Q/R + 12}\)).

Second Most Common Error:

Poor SIMPLIFY strategy: Students attempt to manipulate the original equation incorrectly.

Instead of dividing by R first, they might try to subtract 12 from both sides of \(\mathrm{Q = R(S + 12)}\), getting \(\mathrm{Q - 12 = R(S + 12) - 12 = RS}\). Then they divide by R to get \(\mathrm{S = (Q - 12)/R}\). This leads them to select Choice D (\(\mathrm{S = (Q - 12)/R}\)).

The Bottom Line:

This problem tests systematic algebraic manipulation. Success requires following the proper sequence: first eliminate the coefficient (divide by R), then eliminate the constant term (subtract 12). Rushing or changing the order creates sign errors and wrong expressions.