Is 5/9 Bigger Than 11/12?

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 \(\frac{11}{12}\): multiply numerator and denominator by 3.
  \[\frac{11}{12} = \frac{11 \times 3}{12 \times 3} = \frac{33}{36}\]
• Step 3. Compare the numerators: 20 vs 33.
• Step 4. Since 20 is less than 33, we know that \(\frac{5}{9} < \frac{11}{12}\).

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: \(5 \times 12 = 60\).
• Step 2. Multiply the numerator of the second fraction by the denominator of the first: \(11 \times 9 = 99\).
• Step 3. Compare the products: \(60 < 99\), so \(\frac{5}{9} < \frac{11}{12}\).

Method 3: Decimal Comparison

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

Since 0.555556 is less than 0.916667, we confirm that \(\frac{5}{9} < \frac{11}{12}\). In percentage terms, \(\frac{5}{9}\) is 55.5556% and \(\frac{11}{12}\) is 91.6667%, a difference of 36.1111 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 36, we're cutting both quantities into equal-sized pieces. Then 20 pieces vs 33 pieces is a straightforward comparison.