If \(6(\mathrm{y} - 5) = 24\), what is the value of \(2(\mathrm{y} - 5)\)? 2 4 8 12...

1. INFER the most efficient approach

• Given: \(6(\mathrm{y} - 5) = 24\)
• Find: \(2(\mathrm{y} - 5)\)

Key insight: I don't need to solve for y! I can treat \((\mathrm{y} - 5)\) as a single unit and work with it directly.

2. SIMPLIFY to find the value of \((\mathrm{y} - 5)\)

• Divide both sides by 6:

\(6(\mathrm{y} - 5) = 24\)
\((\mathrm{y} - 5) = 24 ÷ 6 = 4\)

3. SIMPLIFY to find \(2(\mathrm{y} - 5)\)

• Now that I know \((\mathrm{y} - 5) = 4\):

\(2(\mathrm{y} - 5) = 2(4) = 8\)

Answer: C (8)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize they can work with \((\mathrm{y} - 5)\) as a unit and instead expand the original equation unnecessarily.

They might distribute: \(6(\mathrm{y} - 5) = 24\) → \(6\mathrm{y} - 30 = 24\) → \(6\mathrm{y} = 54\) → \(\mathrm{y} = 9\), then substitute back: \(2(9 - 5) = 2(4) = 8\). While this approach still yields the correct answer, it involves more steps and increases chances for arithmetic errors.

This leads to taking longer on the problem and potential calculation mistakes along the way.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when dividing 24 by 6 or when calculating the final multiplication.

For example, if they incorrectly calculate \(24 ÷ 6 = 3\), they would get \(2(\mathrm{y} - 5) = 2(3) = 6\), leading them to select Choice B (4) (if they then make another error) or become confused about which answer choice matches their result.

The Bottom Line:

The elegant solution recognizes that expressions can be treated as single units in algebraic manipulation. This problem rewards students who can see the direct relationship between \(6(\mathrm{y} - 5)\) and \(2(\mathrm{y} - 5)\) without unnecessarily solving for individual variables.