If 8x = 6, what is the value of 72x?

1. TRANSLATE the problem information

• Given: \(\mathrm{8x = 6}\)
• Find: The value of \(\mathrm{72x}\)

2. INFER the most efficient approach

• Instead of solving for x first, notice that \(\mathrm{72x}\) is related to \(\mathrm{8x}\)
• Key insight: \(\mathrm{72x = 9 \times 8x}\) (since \(\mathrm{72 \div 8 = 9}\))
• This means I can use the given equation directly

3. SIMPLIFY using substitution

• Since \(\mathrm{8x = 6}\), then \(\mathrm{9 \times 8x = 9 \times 6}\)
• Therefore: \(\mathrm{72x = 9 \times 6 = 54}\)

Answer: C. 54

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the relationship between \(\mathrm{8x}\) and \(\mathrm{72x}\), so they solve for x first: \(\mathrm{x = \frac{6}{8} = \frac{3}{4}}\), then calculate \(\mathrm{72x = 72 \times \frac{3}{4}}\). While this approach works, it's more prone to arithmetic errors with fractions.

Students might make calculation errors when working with \(\mathrm{72 \times \frac{3}{4}}\), potentially getting confused with the fraction arithmetic and selecting an incorrect answer.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misread the problem and think they need to find the value of x rather than \(\mathrm{72x}\). They correctly calculate \(\mathrm{x = \frac{3}{4}}\) but then look for this value among the choices. Since \(\mathrm{\frac{3}{4}}\) isn't listed, they might select Choice A (3), which is actually the value of \(\mathrm{4x}\).

The Bottom Line:

This problem rewards students who can see mathematical relationships rather than just following mechanical procedures. The key insight is recognizing that you can work with the given equation directly without solving for the variable first.