Is 1/2 Bigger Than 2/3?

Method 1: Common Denominators

To compare these fractions, we need a common denominator. The denominators are 2 and 3, and the least common denominator (LCD) is 6.

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

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: \(1 \times 3 = 3\).
• Step 2. Multiply the numerator of the second fraction by the denominator of the first: \(2 \times 2 = 4\).
• Step 3. Compare the products: \(3 < 4\), so \(\frac{1}{2} < \frac{2}{3}\).

Method 3: Decimal Comparison

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

Since 0.5 is less than 0.666667, we confirm that \(\frac{1}{2} < \frac{2}{3}\). In percentage terms, \(\frac{1}{2}\) is 50% and \(\frac{2}{3}\) is 66.6667%, 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 3 pieces vs 4 pieces is a straightforward comparison.