Is 2/9 Bigger Than 2/11?

Method 1: Common Denominators

To compare these fractions, we need a common denominator. The denominators are 9 and 11, and the least common denominator (LCD) is 99.

• Step 1. Convert \(\frac{2}{9}\): multiply numerator and denominator by 11.
  \[\frac{2}{9} = \frac{2 \times 11}{9 \times 11} = \frac{22}{99}\]
• Step 2. Convert \(\frac{2}{11}\): multiply numerator and denominator by 9.
  \[\frac{2}{11} = \frac{2 \times 9}{11 \times 9} = \frac{18}{99}\]
• Step 3. Compare the numerators: 22 vs 18.
• Step 4. Since 22 is greater than 18, we know that \(\frac{2}{9} > \frac{2}{11}\).

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: \(2 \times 11 = 22\).
• Step 2. Multiply the numerator of the second fraction by the denominator of the first: \(2 \times 9 = 18\).
• Step 3. Compare the products: \(22 > 18\), so \(\frac{2}{9} > \frac{2}{11}\).

Method 3: Decimal Comparison

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

Since 0.222222 is greater than 0.181818, we confirm that \(\frac{2}{9} > \frac{2}{11}\). In percentage terms, \(\frac{2}{9}\) is 22.2222% and \(\frac{2}{11}\) is 18.1818%, a difference of 4.0404 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 11ths are larger pieces than 9ths, \(\frac{2}{9}\) is the bigger fraction even though both have 2 in the numerator.