Is 1/9 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 \(\frac{1}{9}\): multiply numerator and denominator by 11.
  \[\frac{1}{9} = \frac{1 \times 11}{9 \times 11} = \frac{11}{99}\]
• 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: 11 vs 18.
• Step 4. Since 11 is less than 18, we know that \(\frac{1}{9} < \frac{2}{11}\).

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: \(1 \times 11 = 11\).
• Step 2. Multiply the numerator of the second fraction by the denominator of the first: \(2 \times 9 = 18\).
• Step 3. Compare the products: \(11 < 18\), so \(\frac{1}{9} < \frac{2}{11}\).

Method 3: Decimal Comparison

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

Since 0.111111 is less than 0.181818, we confirm that \(\frac{1}{9} < \frac{2}{11}\). In percentage terms, \(\frac{1}{9}\) is 11.1111% and \(\frac{2}{11}\) is 18.1818%, a difference of 7.0707 percentage points.

Why Does This Work?

These fractions have different numerators and different denominators, so we can't compare them directly. By converting to a common denominator of 99, we're cutting both quantities into equal-sized pieces. Then 11 pieces vs 18 pieces is a straightforward comparison.