Which expression is equivalent to (18m + 24)/6?3m + 4\(3(\mathrm{m} + 4)\)(3m + 4)/618m + 4\(6(3\mathrm{m} + 4)\)

1. INFER the best approach

• We have a fraction with an algebraic expression in the numerator: \(\frac{18\mathrm{m} + 24}{6}\)
• We can either divide each term separately or factor first - both work!

2. SIMPLIFY using the distributive property of division

• Split the fraction: \(\frac{18\mathrm{m} + 24}{6} = \frac{18\mathrm{m}}{6} + \frac{24}{6}\)
• Divide each term: \(\frac{18\mathrm{m}}{6} = 3\mathrm{m}\) and \(\frac{24}{6} = 4\)
• Combined result: \(3\mathrm{m} + 4\)

3. Verify with factoring method

• SIMPLIFY by factoring first: \(18\mathrm{m} + 24 = 6(3\mathrm{m} + 4)\)
• SIMPLIFY by canceling: \(\frac{6(3\mathrm{m} + 4)}{6} = 3\mathrm{m} + 4\) ✓

Answer: A (\(3\mathrm{m} + 4\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Incomplete SIMPLIFY execution: Students divide only one term instead of both.
They might think: \(\frac{18\mathrm{m} + 24}{6} = 18\mathrm{m} + \frac{24}{6} = 18\mathrm{m} + 4\), treating division like it only applies to the last term.
This leads them to select Choice D (\(18\mathrm{m} + 4\)).

Second Most Common Error:

Stopping partway through SIMPLIFY process: Students factor correctly but don't complete the simplification.
They get \(18\mathrm{m} + 24 = 6(3\mathrm{m} + 4)\), so \(\frac{18\mathrm{m} + 24}{6} = 6(3\mathrm{m} + 4)\), and think this is the final answer without canceling the 6's.
This may lead them to select Choice E [\(6(3\mathrm{m} + 4)\)].

The Bottom Line:

The key insight is that division must be distributed to every term in the numerator, and any factoring must be followed through to complete cancellation.