Is 1/2 Bigger Than 1/3?

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

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: \(1 \times 2 = 2\).
• Step 3. Compare the products: \(3 > 2\), so \(\frac{1}{2} > \frac{1}{3}\).

Method 3: Decimal Comparison

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

Since 0.5 is greater than 0.333333, we confirm that \(\frac{1}{2} > \frac{1}{3}\). In percentage terms, \(\frac{1}{2}\) is 50% and \(\frac{1}{3}\) is 33.3333%, a difference of 16.6667 percentage points.

Why Does This Work?

When two fractions have the same numerator, you have the same number of pieces — but the pieces are different sizes. A smaller denominator means each piece is larger. Since 3ths are larger pieces than 2ths, \(\frac{1}{2}\) is the bigger fraction even though both have 1 in the numerator.