Question:1/5b = 2/xyThe given equation relates the positive numbers b, x, and y. Which equation correctly expresses x in terms...

1. TRANSLATE the problem information

• Given equation: \(\frac{1}{5\mathrm{b}} = \frac{2}{\mathrm{xy}}\)
• Need to find: x in terms of b and y

2. INFER the solution strategy

• When we have two fractions equal to each other, cross-multiplication is the most efficient approach
• This will eliminate the denominators and give us a simpler equation to work with

3. SIMPLIFY using cross-multiplication

• Cross-multiply: \(1 \times \mathrm{xy} = 2 \times 5\mathrm{b}\)
• This gives us: \(\mathrm{xy} = 10\mathrm{b}\)
• Now we have a much simpler equation without fractions

4. SIMPLIFY to isolate x

• We want x by itself, so divide both sides by y
• \(\mathrm{xy} \div \mathrm{y} = 10\mathrm{b} \div \mathrm{y}\)
• This gives us: \(\mathrm{x} = \frac{10\mathrm{b}}{\mathrm{y}}\)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: After getting \(\mathrm{xy} = 10\mathrm{b}\), students incorrectly move the y to the other side by multiplying instead of dividing.

They think: "To get x alone, I need to get rid of y, so I'll multiply both sides by y." This gives them \(\mathrm{x} = 10\mathrm{by}\), leading them to select Choice A (\(\mathrm{x} = 10\mathrm{by}\)).

Second Most Common Error:

Inadequate INFER reasoning: Students attempt cross-multiplication but don't complete it properly, getting confused about which terms multiply with which.

They might partially cross-multiply as \(1 \times \mathrm{xy} = 2\mathrm{y}\) and \(5\mathrm{b} \times 2\) = (something with 5b), leading to jumbled algebra. This confusion may lead them to select Choice B (\(\mathrm{x} = \frac{2\mathrm{y}}{5\mathrm{b}}\)) or causes them to get stuck and guess.

The Bottom Line:

This problem tests whether students can systematically apply cross-multiplication and then correctly isolate a variable through division, not multiplication.