\(\mathrm{f(x) = (x + 0.25x)(50 - x)}\)The function f is defined above. What is the value of \(\mathrm{f(20)}\)?

1. SIMPLIFY the function first

• Given: \(\mathrm{f(x) = (x + 0.25x)(50 - x)}\)
• Combine like terms in the first parentheses: \(\mathrm{x + 0.25x = 1.25x}\)
• Simplified function: \(\mathrm{f(x) = (1.25x)(50 - x)}\)

2. TRANSLATE the question into mathematical action

• \(\mathrm{f(20)}\) means: substitute 20 for every x in the function
• \(\mathrm{f(20) = (1.25 \times 20)(50 - 20)}\)

3. SIMPLIFY through arithmetic calculation

• Calculate the first part: \(\mathrm{1.25 \times 20 = 25}\)
• Calculate the second part: \(\mathrm{50 - 20 = 30}\)
• Final multiplication: \(\mathrm{f(20) = 25 \times 30 = 750}\)

Answer: C. 750

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make arithmetic errors in the calculations, particularly with decimal multiplication.

Some students correctly set up \(\mathrm{f(20) = (1.25 \times 20)(50 - 20)}\) but then calculate \(\mathrm{1.25 \times 20}\) incorrectly, or make errors in the final multiplication \(\mathrm{25 \times 30}\). These computational mistakes can lead to selecting incorrect answer choices or cause confusion about which answer to pick.

Second Most Common Error:

Incomplete SIMPLIFY execution: Students skip the like terms combination step and work with the original form.

Instead of first combining \(\mathrm{x + 0.25x = 1.25x}\), they try to work directly with \(\mathrm{f(20) = (20 + 0.25 \times 20)(50 - 20)}\). While this approach can still work, it creates more opportunities for arithmetic errors and makes the problem unnecessarily complicated. This often leads to calculation mistakes and wrong answer selection.

The Bottom Line:

This problem tests whether students can efficiently combine like terms before substitution and execute accurate decimal arithmetic. The key insight is recognizing that simplifying first makes the calculation much cleaner and reduces error opportunities.