Is 5/9 Bigger Than 5/11?

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 \(\frac{5}{9}\): multiply numerator and denominator by 11.
  \[\frac{5}{9} = \frac{5 \times 11}{9 \times 11} = \frac{55}{99}\]
• Step 2. Convert \(\frac{5}{11}\): multiply numerator and denominator by 9.
  \[\frac{5}{11} = \frac{5 \times 9}{11 \times 9} = \frac{45}{99}\]
• Step 3. Compare the numerators: 55 vs 45.
• Step 4. Since 55 is greater than 45, we know that \(\frac{5}{9} > \frac{5}{11}\).

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: \(5 \times 11 = 55\).
• Step 2. Multiply the numerator of the second fraction by the denominator of the first: \(5 \times 9 = 45\).
• Step 3. Compare the products: \(55 > 45\), so \(\frac{5}{9} > \frac{5}{11}\).

Method 3: Decimal Comparison

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

Since 0.555556 is greater than 0.454545, we confirm that \(\frac{5}{9} > \frac{5}{11}\). In percentage terms, \(\frac{5}{9}\) is 55.5556% and \(\frac{5}{11}\) is 45.4545%, a difference of 10.101 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, \(\frac{5}{9}\) is the bigger fraction even though both have 5 in the numerator.