If 2n/5 = 10, what is the value of 2n - 1?

1. TRANSLATE the problem information

• Given equation: \(\frac{2n}{5} = 10\)
• Find: The value of \(2n - 1\)
• What this tells us: We have an equation involving \(2n\), and we need to find an expression involving \(2n\)

2. INFER the most efficient approach

• Key insight: Since we need \(2n - 1\), we should find the value of \(2n\) first
• Strategy: Solve for \(2n\) directly rather than finding \(n\) first
• This saves a step and reduces potential for errors

3. SIMPLIFY to find \(2n\)

• Start with: \(\frac{2n}{5} = 10\)
• Multiply both sides by 5: \(2n = 50\)
• Now we know \(2n = 50\)

4. SIMPLIFY to find the final answer

• We need: \(2n - 1\)
• Substitute: \(2n - 1 = 50 - 1 = 49\)

Answer: B. 49

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students solve for \(n\) first instead of recognizing they can find \(2n\) directly

Students think: "I need to find \(n = 25\), then calculate \(2n - 1 = 2(25) - 1\)"

While this works, it introduces an extra step where they might make errors, particularly if they miscalculate \(n\) or forget to multiply by 2 correctly.

Second Most Common Error:

Poor TRANSLATE understanding: Students find the wrong expression value

Students correctly find \(2n = 50\) but then:

• Find just \(2n\) instead of \(2n - 1\), leading to Choice C (50)
• Somehow find \(n - 1 = 24\) instead of \(2n - 1\), leading to Choice A (24)
• Make algebraic errors and find \(4n - 1 = 99\), leading to Choice D (99)

The Bottom Line:

This problem tests whether students can work efficiently with expressions and avoid getting distracted by unnecessary steps. The key insight is recognizing that you don't always need to solve for the variable itself—sometimes you can solve directly for the expression you need.