Under what conditions does the order reverse or become equality in the common numerator comparison strategy?

Conditions

• Comparing fractions $\frac{n}{a}$ and $\frac{n}{b}$ with $0 < a < b$
• Case 1: $n > 0$ (standard case)
• Case 2: $n = 0$
• Case 3: $n < 0$

Reasoning, step by step

1. Analyze Case $n > 0$: $\frac{n}{a} > \frac{n}{b}$ because $a < b$ implies larger unit size for positive counts.
2. Analyze Case $n = 0$: $\frac{0}{a} = 0$ and $\frac{0}{b} = 0$, so $\frac{0}{a} = \frac{0}{b}$.
3. Analyze Case $n < 0$: Let $n = -k$ where $k > 0$. Then $-\frac{k}{a}$ vs $-\frac{k}{b}$. Since $\frac{k}{a} > \frac{k}{b}$, negating flips the inequality to $-\frac{k}{a} < -\frac{k}{b}$.
4. Conclusion: The direction of inequality depends on the sign of the numerator.

Example

The card 'Inverse Relationship with Denominators' specifies: 'A zero numerator gives equality; a negative common numerator reverses the order.' Formula context: $n>0,\ 0<a<b\quad\Rightarrow\quad\frac{n}{a}>\frac{n}{b}$.

Common misconceptions

• Applying the 'smaller denominator is bigger' rule blindly without checking the sign of the numerator.
• Assuming that zero numerators behave like positive ones regarding magnitude comparisons.

Watch the explanation

Explore next

• How do you handle mixed signs in the denominators during this comparison?
• Does the absolute value of the numerator affect the strictness of the inequality?