Question:If 3m/4n = 7, what is the value of 4n/3m?

1. TRANSLATE the problem information

• Given: \(\frac{3\mathrm{m}}{4\mathrm{n}} = 7\)
• Find: \(\frac{4\mathrm{n}}{3\mathrm{m}}\)

2. INFER the key relationship

• Compare what we have \(\frac{3\mathrm{m}}{4\mathrm{n}}\) with what we need \(\frac{4\mathrm{n}}{3\mathrm{m}}\)
• Notice that \(\frac{4\mathrm{n}}{3\mathrm{m}}\) has the numerator and denominator flipped compared to \(\frac{3\mathrm{m}}{4\mathrm{n}}\)
• This means \(\frac{4\mathrm{n}}{3\mathrm{m}}\) is the reciprocal (multiplicative inverse) of \(\frac{3\mathrm{m}}{4\mathrm{n}}\)

3. SIMPLIFY using the reciprocal relationship

• Since \(\frac{3\mathrm{m}}{4\mathrm{n}} = 7\), we have:
• \(\frac{4\mathrm{n}}{3\mathrm{m}} = \frac{1}{\frac{3\mathrm{m}}{4\mathrm{n}}} = \frac{1}{7}\)

Answer: B) \(\frac{1}{7}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the reciprocal relationship between the given expression and what they need to find. Instead, they attempt to solve for the individual variables m and n.

They might try: "If \(\frac{3\mathrm{m}}{4\mathrm{n}} = 7\), then \(3\mathrm{m} = 28\mathrm{n}\), so \(\mathrm{m} = \frac{28\mathrm{n}}{3}\). Then substitute this back..." This leads to unnecessarily complex algebra that doesn't utilize the elegant reciprocal relationship. This approach often leads to confusion and guessing, or students may try to eliminate answer choices that don't seem to fit their complex expressions.

The Bottom Line:

The key insight is recognizing patterns in algebraic expressions. When you see two fractions where the numerators and denominators are swapped, think "reciprocal" - this recognition transforms a potentially complex problem into a one-step solution.