Given \(\frac{(\mathrm{k} - 48)}{\mathrm{m}} = \mathrm{p} + 3\), where k, m, and p are positive numbers, which equation correctly expresses...

1. TRANSLATE the problem information

• Given equation: \(\frac{\mathrm{k - 48}}{\mathrm{m}} = \mathrm{p + 3}\)
• Goal: Express k in terms of m and p

2. INFER the solution strategy

• We need to isolate k on one side of the equation
• The fraction makes this tricky, so we should eliminate it first
• Strategy: Multiply both sides by m to clear the denominator

3. SIMPLIFY by eliminating the fraction

• Multiply both sides by m:

\(\frac{\mathrm{k - 48}}{\mathrm{m}} \times \mathrm{m} = (\mathrm{p + 3}) \times \mathrm{m}\)

• This gives us: \(\mathrm{k - 48 = m(p + 3)}\)

4. SIMPLIFY by applying the distributive property

• Distribute m to both terms in the parentheses:

\(\mathrm{k - 48 = mp + 3m}\)

5. SIMPLIFY to isolate k

• Add 48 to both sides:

\(\mathrm{k - 48 + 48 = mp + 3m + 48}\)

• Final result: \(\mathrm{k = mp + 3m + 48}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students incorrectly apply the distributive property when multiplying \(\mathrm{m(p + 3)}\), treating it as \(\mathrm{mp + 3}\) instead of \(\mathrm{mp + 3m}\).

This fundamental algebra mistake occurs when students forget that the m must be distributed to both terms inside the parentheses. They might think \(\mathrm{m(p + 3) = mp + 3}\), completely missing that the 3 also needs to be multiplied by m.

This error leads them to select Choice A (\(\mathrm{k = mp + 3 + 48}\)).

Second Most Common Error:

Poor SIMPLIFY reasoning: Students make a sign error when moving the 48 to the other side of the equation, subtracting instead of adding.

After correctly getting \(\mathrm{k - 48 = mp + 3m}\), they might think "to get rid of the -48, I subtract 48 from both sides" instead of recognizing they need to add 48 to both sides to isolate k.

This leads them to select Choice D (\(\mathrm{k = mp + 3m - 48}\)).

The Bottom Line:

This problem tests careful algebraic manipulation skills. Success requires methodical application of the distributive property and precise attention to arithmetic signs when moving terms across the equals sign.