If 4x = 3, what is the value of 24x?

1. TRANSLATE the problem information

• Given: \(\mathrm{4x = 3}\)
• Find: \(\mathrm{24x}\) (not x itself)

2. INFER the relationship between given and target expressions

• Key insight: Notice that \(\mathrm{24x}\) is a multiple of \(\mathrm{4x}\)
• Specifically: \(\mathrm{24x = 6 \times 4x}\)
• This means we can use our known value without solving for x

3. SIMPLIFY by substitution

• Since \(\mathrm{24x = 6 \times 4x}\) and \(\mathrm{4x = 3}\):
• \(\mathrm{24x = 6 \times 3 = 18}\)

Answer: C. 18

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the direct relationship between \(\mathrm{4x}\) and \(\mathrm{24x}\), and instead try to solve for x first.

They work: \(\mathrm{4x = 3}\) → \(\mathrm{x = \frac{3}{4}}\) → \(\mathrm{24x = 24\left(\frac{3}{4}\right) = 18}\). While this gives the correct answer, it's unnecessarily complex and creates more opportunities for arithmetic errors.

Second Most Common Error:

Poor INFER reasoning: Students misidentify which multiple they need to calculate.

They might calculate:

• \(\mathrm{6x = 6\left(\frac{3}{4}\right) = \frac{9}{2}}\), leading to Choice A
• \(\mathrm{8x = 2(4x) = 2(3) = 6}\), leading to Choice B
• \(\mathrm{96x = 24(4x) = 24(3) = 72}\), leading to Choice D

The Bottom Line:

The key breakthrough is recognizing that you can work directly with the given equation \(\mathrm{4x = 3}\) without solving for x. Once you see that \(\mathrm{24x = 6(4x)}\), the problem becomes straightforward multiplication.