Is 5/9 Bigger Than 5/12?

Method 1: Common Denominators

To compare these fractions, we need a common denominator. The denominators are 9 and 12, and the least common denominator (LCD) is 36.

• Step 1. Convert \(\frac{5}{9}\): multiply numerator and denominator by 4.
  \[\frac{5}{9} = \frac{5 \times 4}{9 \times 4} = \frac{20}{36}\]
• Step 2. Convert \(\frac{5}{12}\): multiply numerator and denominator by 3.
  \[\frac{5}{12} = \frac{5 \times 3}{12 \times 3} = \frac{15}{36}\]
• Step 3. Compare the numerators: 20 vs 15.
• Step 4. Since 20 is greater than 15, we know that \(\frac{5}{9} > \frac{5}{12}\).

Method 2: Cross Multiplication

A quicker way to compare fractions is to cross-multiply. Multiply each numerator by the other fraction's denominator:

• Step 1. Multiply the numerator of the first fraction by the denominator of the second: \(5 \times 12 = 60\).
• Step 2. Multiply the numerator of the second fraction by the denominator of the first: \(5 \times 9 = 45\).
• Step 3. Compare the products: \(60 > 45\), so \(\frac{5}{9} > \frac{5}{12}\).

Method 3: Decimal Comparison

Convert each fraction to a decimal by dividing the numerator by the denominator:

Since 0.555556 is greater than 0.416667, we confirm that \(\frac{5}{9} > \frac{5}{12}\). In percentage terms, \(\frac{5}{9}\) is 55.5556% and \(\frac{5}{12}\) is 41.6667%, a difference of 13.8889 percentage points.

Why Does This Work?

When two fractions have the same numerator, you have the same number of pieces — but the pieces are different sizes. A smaller denominator means each piece is larger. Since 12ths are larger pieces than 9ths, \(\frac{5}{9}\) is the bigger fraction even though both have 5 in the numerator.