A shipping company models the cost \(\mathrm{F(w)}\), in dollars, to ship a package of weight w kilograms by the function...

1. INFER what we need to find

• We need the slope of \(\mathrm{F(w) = \frac{1}{5}(27w - 10) + \frac{3}{2}w + 8}\)
• For any linear function \(\mathrm{F(w) = (coefficient\ of\ w) \cdot w + constant}\), the slope is the coefficient of w
• Strategy: Simplify the function to clearly identify the w coefficient

2. SIMPLIFY by distributing the fraction

• Distribute \(\mathrm{\frac{1}{5}}\) through the parentheses:
• \(\mathrm{\frac{1}{5}(27w - 10) = \frac{1}{5}(27w) - \frac{1}{5}(10) = \frac{27}{5}w - 2}\)
• Now we have: \(\mathrm{F(w) = \frac{27}{5}w - 2 + \frac{3}{2}w + 8}\)

3. SIMPLIFY by combining like terms

• Identify the w terms: \(\mathrm{\frac{27}{5}w}\) and \(\mathrm{\frac{3}{2}w}\)
• Identify the constant terms: -2 and 8 (which combine to 6)
• We need to add the coefficients: \(\mathrm{\frac{27}{5} + \frac{3}{2}}\)

4. SIMPLIFY the fraction addition

• Find common denominator for \(\mathrm{\frac{27}{5} + \frac{3}{2}}\)
• LCD of 5 and 2 is 10
• Convert: \(\mathrm{\frac{27}{5} = \frac{54}{10}}\) and \(\mathrm{\frac{3}{2} = \frac{15}{10}}\)
• Add: \(\mathrm{\frac{54}{10} + \frac{15}{10} = \frac{69}{10}}\)

Answer: \(\mathrm{\frac{69}{10}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make arithmetic errors when adding fractions with different denominators.

Many students correctly identify that they need \(\mathrm{\frac{27}{5} + \frac{3}{2}}\) but then make calculation errors. They might:

• Use wrong common denominator (like 6 instead of 10)
• Make conversion errors (like \(\mathrm{\frac{27}{5} = \frac{27}{10}}\) instead of \(\mathrm{\frac{54}{10}}\))
• Add numerators and denominators separately (getting something like \(\mathrm{\frac{30}{7}}\))

This leads to confusion and incorrect final answers.

Second Most Common Error:

Poor INFER reasoning: Students don't recognize they need to find the coefficient of w.

Some students might try to substitute values or approach this as if they need to graph the function rather than recognizing this is about the algebraic structure. They get overwhelmed by the complex expression and don't realize the straightforward approach of simplifying to standard form.

This causes them to get stuck and abandon systematic solution.

The Bottom Line:

This problem tests whether students can systematically simplify expressions with fractions while keeping sight of the goal - finding the slope means finding the coefficient of the variable.