b = 42cfThe given equation relates the positive numbers b, c, and f. Which equation correctly expresses c in terms...

1. TRANSLATE the problem information

• Given equation: \(\mathrm{b = 42cf}\)
• Need to find: \(\mathrm{c}\) expressed in terms of \(\mathrm{b}\) and \(\mathrm{f}\)
• What this tells us: \(\mathrm{c}\) is currently multiplied by both \(\mathrm{42}\) and \(\mathrm{f}\)

2. INFER the approach needed

• To isolate \(\mathrm{c}\), we need to "undo" the multiplication by \(\mathrm{42}\) and \(\mathrm{f}\)
• Since \(\mathrm{c}\) is multiplied by \(\mathrm{42f}\), we divide both sides by \(\mathrm{42f}\)
• This uses the property that multiplication and division are inverse operations

3. SIMPLIFY by performing the division

• Divide both sides by \(\mathrm{42f}\):
  \(\mathrm{b \div (42f) = (42cf) \div (42f)}\)
• The right side simplifies: \(\mathrm{\frac{42cf}{42f} = c}\)
• This gives us: \(\mathrm{\frac{b}{42f} = c}\)

4. Rewrite in standard form

• \(\mathrm{c = \frac{b}{42f}}\)
• This can also be written as \(\mathrm{c = \frac{b}{42f}}\)

Answer: A. \(\mathrm{c = \frac{b}{42f}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that division is needed to isolate a variable from multiplication.

Students often think about "moving terms to the other side" using addition/subtraction, even when the relationship is multiplicative. They might see \(\mathrm{b = 42cf}\) and think "I need to get rid of the 42, so I'll subtract it from both sides," leading to \(\mathrm{b - 42 = cf}\), then dividing by \(\mathrm{f}\) to get \(\mathrm{c = \frac{b - 42}{f}}\).

This may lead them to select Choice B \(\mathrm{\frac{b-42}{f}}\).

Second Most Common Error:

Poor SIMPLIFY execution: Confusing which operation to perform or making arithmetic errors.

Some students recognize they need to manipulate the equation but get confused about whether to multiply or divide. They might think "I have \(\mathrm{b = 42cf}\) and I want \(\mathrm{c}\), so I need to involve all these terms" and incorrectly multiply: \(\mathrm{c = 42bf}\).

This may lead them to select Choice C \(\mathrm{42bf}\).

The Bottom Line:

The key insight is recognizing that when a variable is part of a product (like \(\mathrm{c}\) in \(\mathrm{42cf}\)), you isolate it by dividing by everything else it's multiplied with. This is fundamentally different from equations involving addition or subtraction.