Why does the fraction with the smaller denominator have a greater value when numerators are identical and positive?

Conditions

• Numerators are identical and positive ($n > 0$)
• Denominators are positive integers

Reasoning, step by step

1. Fix the numerator count (e.g., 8 successful shots or 8 pizza slices).
2. Vary the denominator (total attempts or total slices).
3. Observe that dividing a fixed quantity into fewer groups yields larger group sizes.
4. Mathematically, for $a < b$, $\frac{1}{a} > \frac{1}{b}$.
5. Multiply by the common numerator $n$: $\frac{n}{a} > \frac{n}{b}$.

Example

The script uses analogies: 'With eight successful shots, seventeen attempts give a higher success rate than eighteen' and 'taking eight of seventeen pieces gives more than taking eight of eighteen.'

Common misconceptions

• Believing that a larger denominator inherently increases the fraction's value regardless of the numerator.
• Confusing the relationship between division and multiplication when analyzing inverse proportions.

Watch the explanation

Explore next

• What happens to the inequality if the common numerator is negative?
• Is this rule valid if the numerator is zero?