If p + 4 = (2q - 3)/5, what is q in terms of p?

1. INFER the solution strategy

• Given: \(\mathrm{p + 4 = \frac{2q - 3}{5}}\)
• Goal: Express q in terms of p
• Strategy: Use inverse operations to systematically isolate q

2. SIMPLIFY by eliminating the fraction first

• Multiply both sides by 5:

\(\mathrm{5(p + 4) = 5 \times \frac{2q - 3}{5}}\)

\(\mathrm{5(p + 4) = 2q - 3}\)

3. SIMPLIFY by expanding the left side

• Apply distributive property:

\(\mathrm{5p + 20 = 2q - 3}\)

4. SIMPLIFY by collecting constant terms

• Add 3 to both sides:

\(\mathrm{5p + 20 + 3 = 2q - 3 + 3}\)

\(\mathrm{5p + 23 = 2q}\)

5. SIMPLIFY by isolating q

• Divide both sides by 2:

\(\mathrm{q = \frac{5p + 23}{2}}\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors during the multi-step algebraic manipulation, particularly when expanding \(\mathrm{5(p + 4)}\) or when combining constants.

For example, they might incorrectly expand to get \(\mathrm{5p + 16}\) instead of \(\mathrm{5p + 20}\), or forget to add 3 to both sides properly. These computational mistakes lead to expressions like \(\mathrm{\frac{5p + 17}{2}}\) instead of \(\mathrm{\frac{5p + 23}{2}}\).

This may lead them to select Choice D (\(\mathrm{\frac{5p + 17}{2}}\)).

Second Most Common Error:

Poor INFER reasoning: Students attempt to solve for q without first eliminating the fraction, leading to messy algebraic manipulation and confusion about the proper sequence of operations.

They might try to work directly with the fraction or make incorrect assumptions about how to handle the relationship between p and q.

This leads to confusion and guessing among the remaining answer choices.

The Bottom Line:

This problem tests systematic algebraic manipulation skills. Success requires careful attention to arithmetic details while following a logical sequence of inverse operations to isolate the target variable.