Question:For the function \(\mathrm{g(x) = \frac{k}{x} + 15}\), k is a constant and \(\mathrm{g(8) = 20}\). What is the value...

1. TRANSLATE the given information into equations

• Given information:
  • Function: \(\mathrm{g(x) = \frac{k}{x} + 15}\)
  • Condition: \(\mathrm{g(8) = 20}\)
  • Find: \(\mathrm{g(5)}\)

• What this tells us: We need to find the value of constant k first, then use it to evaluate \(\mathrm{g(5)}\)

2. INFER the solution strategy

• Since \(\mathrm{g(5)}\) contains the unknown constant k, we can't evaluate it directly
• The key insight: Use the given condition \(\mathrm{g(8) = 20}\) to find k first
• Once we know k, we can substitute it into \(\mathrm{g(5)}\)

3. TRANSLATE the condition \(\mathrm{g(8) = 20}\) into an equation

Substitute x = 8 into the function:

\(\mathrm{g(8) = \frac{k}{8} + 15 = 20}\)

4. SIMPLIFY to solve for k

• Subtract 15 from both sides: \(\mathrm{\frac{k}{8} = 20 - 15 = 5}\)
• Multiply both sides by 8: \(\mathrm{k = 8 \times 5 = 40}\)

5. SIMPLIFY to find \(\mathrm{g(5)}\)

Now substitute k = 40 and x = 5:

\(\mathrm{g(5) = \frac{40}{5} + 15 = 8 + 15 = 23}\)

Answer: 23

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students don't properly convert \(\mathrm{g(8) = 20}\) into the equation \(\mathrm{\frac{k}{8} + 15 = 20}\)

Students might write something like "8 = k/8 + 15" or get confused about which variable to substitute where. This fundamental translation error prevents them from setting up the correct equation to solve for k, leading to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when solving for k or evaluating the final expression

Common mistakes include:

• Solving \(\mathrm{\frac{k}{8} = 5}\) incorrectly (getting k = 5/8 instead of k = 40)
• Computing 40/5 + 15 as 40/(5+15) = 40/20 = 2
• Making sign errors during algebraic manipulation

These computational errors lead to incorrect values that don't match the answer.

The Bottom Line:

This problem tests whether students can work backwards from a function value to find an unknown constant, then use that constant to evaluate the function at a different point. Success requires both accurate equation setup and careful arithmetic execution.