The function g is defined by \(\mathrm{g(x) = 5x - 10}\). If \(\mathrm{g(t)/2 = g(4) + 5}\), where t is...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{g(x) = 5x - 10}\)
  • Equation: \(\mathrm{\frac{g(t)}{2} = g(4) + 5}\)
  • Need to find: value of t

2. INFER the solving strategy

• Strategy: We need to work with known values first, then unknown values
• First find g(4) since we can calculate this directly
• Then use the equation to find g(t)
• Finally solve for t

3. SIMPLIFY to find g(4)

\(\mathrm{g(4) = 5(4) - 10}\)
\(\mathrm{= 20 - 10}\)
\(\mathrm{= 10}\)

4. TRANSLATE the given equation using our result

• Original equation: \(\mathrm{\frac{g(t)}{2} = g(4) + 5}\)
• Substitute \(\mathrm{g(4) = 10}\): \(\mathrm{\frac{g(t)}{2} = 10 + 5 = 15}\)
• Therefore: \(\mathrm{g(t) = 30}\)

5. INFER how to find t

• Since \(\mathrm{g(t) = 30}\) and \(\mathrm{g(x) = 5x - 10}\)
• We need to solve: \(\mathrm{5t - 10 = 30}\)

6. SIMPLIFY the linear equation

\(\mathrm{5t - 10 = 30}\)
\(\mathrm{5t = 40}\)
\(\mathrm{t = 8}\)

Answer: D. 8

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students try to substitute t directly into the original equation \(\mathrm{\frac{g(t)}{2} = g(4) + 5}\) without first finding numerical values.

They might write something like: \(\mathrm{\frac{5t - 10}{2} = (5(4) - 10) + 5}\) and then get confused trying to solve this more complex equation. While this approach can work, it's more prone to algebraic errors and students often make mistakes in the fraction manipulation or distribution.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify that \(\mathrm{g(4) = 10}\) and \(\mathrm{g(t) = 30}\), but make arithmetic errors when solving \(\mathrm{5t - 10 = 30}\).

Common mistakes include forgetting to add 10 to both sides (getting \(\mathrm{5t = 30}\), so \(\mathrm{t = 6}\)) or making sign errors in the algebra.

This may lead them to select Choice B (6).

The Bottom Line:

This problem tests whether students can work systematically through function evaluation and equation solving. The key insight is recognizing that you should evaluate known quantities first, then use those results to find unknown quantities.