The function g is defined by \(\mathrm{g(t) = \frac{5}{4}(t - 3)}\). For what value of t is \(\mathrm{g(t)}\) equal to...

1. TRANSLATE the problem information

• Given information:
  • Function definition: \(\mathrm{g(t) = \frac{5}{4}(t - 3)}\)
  • We need \(\mathrm{g(t) = 10}\)
• What this tells us: We need to set up an equation where the function output equals 10

2. TRANSLATE the question into a mathematical equation

• The question "For what value of t is g(t) equal to 10?" becomes:
  \(\mathrm{g(t) = 10}\)
• Substituting our function: \(\mathrm{\frac{5}{4}(t - 3) = 10}\)

3. SIMPLIFY by isolating the (t - 3) term

• Multiply both sides by 4/5 (the reciprocal of 5/4):
  \(\mathrm{\frac{4}{5} \times [\frac{5}{4}(t - 3)] = 10 \times \frac{4}{5}}\)
• This gives us: \(\mathrm{t - 3 = 8}\)

4. SIMPLIFY by solving for t

• Add 3 to both sides: \(\mathrm{t = 8 + 3 = 11}\)

5. Verify the answer

• Check:
  \(\mathrm{g(11) = \frac{5}{4}(11 - 3)}\)
  \(\mathrm{= \frac{5}{4}(8)}\)
  \(\mathrm{= 10}\) ✓

Answer: C) 11

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may misinterpret the function notation or set up the wrong equation. Some students might try to substitute the answer choices back into the original function rather than setting \(\mathrm{g(t) = 10}\) first. Others might confuse the setup and write something like "\(\mathrm{t = \frac{5}{4}(10 - 3)}\)" instead of "\(\mathrm{\frac{5}{4}(t - 3) = 10}\)."

This leads to confusion and random guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{\frac{5}{4}(t - 3) = 10}\) but make arithmetic errors when multiplying by the reciprocal. A common mistake is calculating \(\mathrm{10 \times \frac{4}{5}}\) incorrectly, perhaps getting \(\mathrm{10 \times 4 \div 5 = 40 \div 5 = 8}\) wrong, or forgetting to add 3 at the end.

This may lead them to select Choice A (5) if they get \(\mathrm{t - 3 = 2}\), or other incorrect choices based on their calculation errors.

The Bottom Line:

This problem tests whether students can translate a function equation setup and execute multi-step algebraic manipulation accurately. The key insight is recognizing that "g(t) equals 10" means you substitute the function expression for g(t) and set it equal to 10.