The function p is defined by \(\mathrm{p(t) = \frac{18}{5t + 2}}\).What is the value of \(\mathrm{p(\frac{1}{2})}\)?Express your answer as an...

1. TRANSLATE the problem information

• Given: \(\mathrm{p(t) = \frac{18}{5t + 2}}\)
• Find: \(\mathrm{p(\frac{1}{2})}\)
• This means substitute \(\mathrm{t = \frac{1}{2}}\) into the function

2. SIMPLIFY by substituting the value

• Replace t with 1/2: \(\mathrm{p(\frac{1}{2}) = \frac{18}{5(\frac{1}{2}) + 2}}\)
• Work inside the denominator first (order of operations)

3. SIMPLIFY the denominator step by step

• Calculate \(\mathrm{5(\frac{1}{2}) = \frac{5}{2}}\)
• Add: \(\mathrm{\frac{5}{2} + 2 = \frac{5}{2} + \frac{4}{2} = \frac{9}{2}}\)
• So we have: \(\mathrm{p(\frac{1}{2}) = \frac{18}{\frac{9}{2}}}\)

4. SIMPLIFY the division by a fraction

• Use the rule: dividing by a fraction = multiplying by its reciprocal
• \(\mathrm{18 \div \frac{9}{2} = 18 \times \frac{2}{9}}\)
• Calculate: \(\mathrm{18 \times 2 = 36}\), then \(\mathrm{36 \div 9 = 4}\)

Answer: 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution with fraction addition: Students often struggle when adding 5/2 + 2, forgetting to convert 2 to the fraction 4/2 first.

They might incorrectly compute \(\mathrm{\frac{5}{2} + 2 = \frac{7}{2}}\), leading to \(\mathrm{p(\frac{1}{2}) = \frac{18}{\frac{7}{2}} = \frac{36}{7} \approx 5.14}\). Since this is a fill-in problem requiring an integer, this leads to confusion and guessing.

Second Most Common Error:

Inadequate SIMPLIFY execution with fraction division: Students remember that dividing by fractions involves reciprocals, but execute it incorrectly.

They might write \(\mathrm{18 \div \frac{9}{2} = 18 \div 9 \times 2 = 2 \times 2 = 4}\), getting the right answer by accident, or worse, \(\mathrm{18 \div 9 \div 2 = 1}\), leading them to be confused when the answer doesn't seem reasonable.

The Bottom Line:

This problem tests whether students can carefully execute fraction arithmetic within function evaluation. The conceptual understanding is straightforward, but the multi-step fraction operations create many opportunities for computational errors.