Comparing fractions

Comparing fractions is an essential skill for understanding rational numbers, but it can feel daunting at first. In reality, the process is often straightforward and boils down to a few simple steps.

Let’s break it down, addressing the key scenarios and applying the rules with clarity and confidence.

Case 1: Same Denominator

When two fractions share the same denominator, comparing them is easy. Simply look at the numerators: the fraction with the larger numerator is the greater one.

Example

Compare $ \frac{3}{2} $ and $ \frac{7}{2} $:

Both fractions have a denominator of $ 2 $.

Since $ 3 < 7 $, it follows that:

$$ \frac{3}{2} < \frac{7}{2} $$

Case 2: Different Denominators

When fractions have different denominators, there are two main methods to compare them:

Finding a common denominator
Rewrite the fractions with the same denominator by multiplying both the numerator and denominator of each fraction by the denominator of the other. This uses the invariance property.
Example: Compare $ \frac{3}{2} $ and $ \frac{4}{7} $.
Multiply the numerator and denominator of the first fraction by $ 7 $:
$$ \frac{3}{2} = \frac{3 \cdot 7}{2 \cdot 7} = \frac{21}{14} $$ Then, multiply the numerator and denominator of the second fraction by $ 2 $: $$ \frac{4}{7} = \frac{4 \cdot 2}{7 \cdot 2} = \frac{8}{14} $$ Now, the equivalent fractions are $ \frac{21}{14} $ and $ \frac{8}{14} $. Comparing the numerators, $ 21 > 8 $, so: $$ \frac{3}{2} > \frac{4}{7} $$
Cross-multiplication
A quicker method is to compare fractions using cross-products. For $ \frac{a}{b} $ and $ \frac{c}{d} $, compare $ a \cdot d $ and $ b \cdot c $:
- If $ a \cdot d < b \cdot c $, then $ \frac{a}{b} < \frac{c}{d} $.
- If $ a \cdot d > b \cdot c $, then $ \frac{a}{b} > \frac{c}{d} $.
- If $ a \cdot d = b \cdot c $, the fractions are equal.
Example: Compare $ \frac{3}{2} $ and $ \frac{4}{7} $. Calculate the cross-products: $$ 3 \cdot 7 = 21 \quad \text{and} \quad 2 \cdot 4 = 8 $$ Since $ 21 > 8 $, we conclude that: $$ \frac{3}{2} > \frac{4}{7} $$

Here are some helpful tips.

Sometimes, fractions can be compared without finding a common denominator or using cross-products:

Proper vs. improper fractions
An improper fraction (numerator greater than the denominator) is always larger than a proper fraction (numerator smaller than the denominator).
Example: Compare $ \frac{3}{2} $ and $ \frac{4}{7} $. The first is improper ($ 3 > 2 $) and the second is proper ($ 4 < 7 $), so without any further calculations, we can conclude: $$ \frac{3}{2} > \frac{4}{7} $$
Same numerator
If the fractions share the same numerator, the one with the smaller denominator is larger because the unit is divided into fewer parts.
Example: Compare $ \frac{5}{6} $ and $ \frac{5}{7} $. Since $ 6 < 7 $, we conclude: $$ \frac{5}{6} > \frac{5}{7} $$

Comparing fractions is ultimately an exercise in logic and careful reasoning.

By following these rules and choosing the method that best fits the situation, you can avoid common mistakes.

However, the real key is to always think about the context: simplify where possible and use the most direct method to compare numbers.

To summarize:

Same denominator? Compare the numerators.
Different denominators? Find a common denominator or use cross-products.
Same numerator? Compare the denominators.
One fraction is proper, and the other is improper? The improper fraction is always larger.

With practice, these techniques will become second nature, allowing you to handle any comparison with confidence.