Which expression is equivalent to 45r/s - 45rs, where s gt 0?

1. INFER the best approach

• Looking at \(\frac{45r}{s} - 45rs\), I notice both terms share the factor \(45r\)
• Strategic insight: Factor out the common term first to simplify the expression

2. SIMPLIFY by factoring out the common term

• Factor out \(45r\): \(45r(\frac{1}{s} - s)\)
• Now I need to simplify what's in the parentheses

3. SIMPLIFY the expression in parentheses

• I have: \(\frac{1}{s} - s\)
• To subtract these fractions, I need a common denominator of \(s\)
• Rewrite \(s\) as \(\frac{s²}{s}\): \(\frac{1}{s} - \frac{s²}{s}\)
• Combine: \(\frac{1 - s²}{s}\)

4. Write the final expression

• \(45r(\frac{1}{s} - s) = \frac{45r(1 - s²)}{s}\)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that factoring out \(45r\) first is the key strategic step. Instead, they try to work directly with the original fractions or attempt to combine terms incorrectly.

This leads to confusion about how to proceed systematically, causing them to abandon the algebraic approach and guess among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly factor out \(45r\) but make errors when combining \(\frac{1}{s} - s\). They might forget to convert \(s\) to \(\frac{s²}{s}\) when finding the common denominator, leading to \(\frac{45r(1-s)}{s}\) instead of the correct \(\frac{45r(1-s²)}{s}\).

This may lead them to select Choice C (\(\frac{45r(1-s)}{s}\)).

The Bottom Line:

This problem tests whether students can systematically factor expressions and work confidently with algebraic fractions. The key insight is recognizing that factoring first makes the fraction arithmetic much more manageable.