Is 2/3 Bigger Than 5/6?

Method 1: Common Denominators

To compare these fractions, we need a common denominator. The denominators are 3 and 6, and the least common denominator (LCD) is 6.

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

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: \(2 \times 6 = 12\).
• Step 2. Multiply the numerator of the second fraction by the denominator of the first: \(5 \times 3 = 15\).
• Step 3. Compare the products: \(12 < 15\), so \(\frac{2}{3} < \frac{5}{6}\).

Method 3: Decimal Comparison

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

Since 0.666667 is less than 0.833333, we confirm that \(\frac{2}{3} < \frac{5}{6}\). In percentage terms, \(\frac{2}{3}\) is 66.6667% and \(\frac{5}{6}\) is 83.3333%, a difference of 16.6667 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 6, we're cutting both quantities into equal-sized pieces. Then 4 pieces vs 5 pieces is a straightforward comparison.