If 6/7p + 18 = 54, what is the value of 7p?

1. TRANSLATE the problem information

• Given equation: \(\frac{6}{7}\mathrm{p} + 18 = 54\)
• We need to find: \(7\mathrm{p}\) (not just p)

2. INFER the solution strategy

• To find \(7\mathrm{p}\), we first need to find p
• Then multiply that result by 7
• Start by isolating the term with p

3. SIMPLIFY to isolate the p term

• Subtract 18 from both sides:

\(\frac{6}{7}\mathrm{p} + 18 = 54\)

\(\frac{6}{7}\mathrm{p} = 36\)

4. SIMPLIFY to solve for p

• To eliminate the fraction \(\frac{6}{7}\), multiply both sides by its reciprocal \(\frac{7}{6}\):

\(\frac{6}{7}\mathrm{p} = 36\)

\(\mathrm{p} = 36 \times \frac{7}{6}\)

\(\mathrm{p} = \frac{36 \times 7}{6} = \frac{252}{6} = 42\)

5. SIMPLIFY to find the final answer

• Now multiply by 7 to get \(7\mathrm{p}\):

\(7\mathrm{p} = 7 \times 42 = 294\)

Answer: 294

Why Students Usually Falter on This Problem

Most Common Error Path:

Poor TRANSLATE reasoning: Students solve for p correctly (\(\mathrm{p} = 42\)) but then submit 42 as their final answer, forgetting that the question specifically asks for the value of \(7\mathrm{p}\).

This leads them to select an incorrect answer of 42 instead of 294, or causes confusion if 42 isn't among the answer choices.

Second Most Common Error:

Weak SIMPLIFY execution: Students make arithmetic errors when multiplying by the reciprocal, such as calculating \(36 \times \frac{7}{6}\) incorrectly. They might get confused about whether to multiply 36 by 7 first or divide by 6 first.

This leads to getting an incorrect value for p, which then carries through to an incorrect final answer for \(7\mathrm{p}\).

The Bottom Line:

This problem tests careful reading (noticing the question asks for \(7\mathrm{p}\)) and systematic algebraic manipulation. The fractional coefficient makes the algebra slightly more complex, but the real challenge is remembering what the question actually asks for after doing all that algebra work.