Is 2/3 Bigger Than 5/9?

Method 1: Common Denominators

To compare these fractions, we need a common denominator. The denominators are 3 and 9, and the least common denominator (LCD) is 9.

• Step 1. Convert \(\frac{2}{3}\): multiply numerator and denominator by 3.
  \[\frac{2}{3} = \frac{2 \times 3}{3 \times 3} = \frac{6}{9}\]
• Step 2. Keep \(\frac{5}{9}\) as it is — the denominator is already 9.
• Step 3. Compare the numerators: 6 vs 5.
• Step 4. Since 6 is greater than 5, we know that \(\frac{2}{3} > \frac{5}{9}\).

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: \(2 \times 9 = 18\).
• Step 2. Multiply the numerator of the second fraction by the denominator of the first: \(5 \times 3 = 15\).
• Step 3. Compare the products: \(18 > 15\), so \(\frac{2}{3} > \frac{5}{9}\).

Method 3: Decimal Comparison

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

Since 0.666667 is greater than 0.555556, we confirm that \(\frac{2}{3} > \frac{5}{9}\). In percentage terms, \(\frac{2}{3}\) is 66.6667% and \(\frac{5}{9}\) is 55.5556%, a difference of 11.1111 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 9, we're cutting both quantities into equal-sized pieces. Then 6 pieces vs 5 pieces is a straightforward comparison.