If 5x = 20, what is the value of 15x?

1. TRANSLATE the problem information

• Given information: \(\mathrm{5x = 20}\)
• What we need to find: the value of \(\mathrm{15x}\)

2. INFER the most efficient approach

• Key insight: Notice that \(\mathrm{15x = 3 \times (5x)}\)
• This means we can multiply both sides of our given equation by 3
• Alternative: We could solve for x first, then multiply by 15

3. SIMPLIFY using the direct method

• Start with: \(\mathrm{5x = 20}\)
• Multiply both sides by 3: \(\mathrm{3(5x) = 3(20)}\)
• This gives us: \(\mathrm{15x = 60}\)

Answer: D. 60

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the relationship between \(\mathrm{5x}\) and \(\mathrm{15x}\), instead getting bogged down trying to solve for x first and potentially making calculation errors along the way.

They might correctly find \(\mathrm{x = 4}\), but then make an arithmetic error when computing \(\mathrm{15 \times 4}\), or they might solve \(\mathrm{5x = 20}\) incorrectly in the first place. This leads to confusion and potentially guessing among the answer choices.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misunderstand what the problem is asking for and think they need to find just x rather than \(\mathrm{15x}\).

Since \(\mathrm{x = 4}\) doesn't appear among the answer choices, this causes them to get stuck and guess randomly, or they might try to manipulate their answer in ways that don't make mathematical sense.

The Bottom Line:

This problem rewards students who can see the multiplicative relationship between coefficients (\(\mathrm{5x}\) and \(\mathrm{15x}\) differ by a factor of 3) rather than those who automatically default to solving for the variable first.