Is 9/11 Bigger Than 11/12?

Method 1: Common Denominators

To compare these fractions, we need a common denominator. The denominators are 11 and 12, and the least common denominator (LCD) is 132.

• Step 1. Convert \(\frac{9}{11}\): multiply numerator and denominator by 12.
  \[\frac{9}{11} = \frac{9 \times 12}{11 \times 12} = \frac{108}{132}\]
• Step 2. Convert \(\frac{11}{12}\): multiply numerator and denominator by 11.
  \[\frac{11}{12} = \frac{11 \times 11}{12 \times 11} = \frac{121}{132}\]
• Step 3. Compare the numerators: 108 vs 121.
• Step 4. Since 108 is less than 121, we know that \(\frac{9}{11} < \frac{11}{12}\).

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: \(9 \times 12 = 108\).
• Step 2. Multiply the numerator of the second fraction by the denominator of the first: \(11 \times 11 = 121\).
• Step 3. Compare the products: \(108 < 121\), so \(\frac{9}{11} < \frac{11}{12}\).

Method 3: Decimal Comparison

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

Since 0.818182 is less than 0.916667, we confirm that \(\frac{9}{11} < \frac{11}{12}\). In percentage terms, \(\frac{9}{11}\) is 81.8182% and \(\frac{11}{12}\) is 91.6667%, a difference of 9.8485 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 132, we're cutting both quantities into equal-sized pieces. Then 108 pieces vs 121 pieces is a straightforward comparison.