The given equation relates the positive numbers a, b, and x. Which equation correctly expresses a in terms of b...

1. TRANSLATE the problem information

• Given equation: \(\mathrm{x = 8a(b + 9)}\)
• Goal: Express a in terms of b and x (isolate a)

2. INFER the solution strategy

• To isolate a, I need to "undo" what's being done to it
• Currently a is being multiplied by \(\mathrm{8(b + 9)}\)
• To undo multiplication, I divide both sides by \(\mathrm{8(b + 9)}\)

3. SIMPLIFY by dividing both sides

• Divide both sides by \(\mathrm{8(b + 9)}\):

\(\frac{\mathrm{x}}{\mathrm{8(b + 9)}} = \frac{\mathrm{8a(b + 9)}}{\mathrm{8(b + 9)}}\)

• The right side simplifies to just a:

\(\frac{\mathrm{x}}{\mathrm{8(b + 9)}} = \mathrm{a}\)

• Rearrange: \(\mathrm{a = \frac{x}{8(b + 9)}}\)

4. TRANSLATE back to match answer format

• My result: \(\mathrm{a = \frac{x}{8(b + 9)}}\)
• This matches choice B: \(\mathrm{a = \frac{x}{8(b+9)}}\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students try to "reverse" operations incorrectly, thinking that if \(\mathrm{x = 8a(b + 9)}\), then \(\mathrm{a = \frac{x}{8} - (b + 9)}\). They incorrectly treat the multiplication by 8 and the addition of (b + 9) as separate operations that need to be undone individually.

This leads them to select Choice A (\(\mathrm{a = \frac{x}{8} - (b + 9)}\))

Second Most Common Error:

Poor TRANSLATE reasoning: Students get confused about fraction notation and think that to solve \(\mathrm{x = 8a(b + 9)}\), they need to "flip" everything, leading them to write \(\mathrm{a = \frac{8(b + 9)}{x}}\).

This may lead them to select Choice C (\(\mathrm{a = \frac{8(b+9)}{x}}\))

The Bottom Line:

The key insight is recognizing that a is multiplied by the entire expression \(\mathrm{8(b + 9)}\), so you must divide by that entire expression to isolate a. Students who try to undo operations piece by piece rather than seeing the complete coefficient will struggle with this type of problem.