Is 1/2 Bigger Than 8/11?

Method 1: Common Denominators

To compare these fractions, we need a common denominator. The denominators are 2 and 11, and the least common denominator (LCD) is 22.

• Step 1. Convert \(\frac{1}{2}\): multiply numerator and denominator by 11.
  \[\frac{1}{2} = \frac{1 \times 11}{2 \times 11} = \frac{11}{22}\]
• Step 2. Convert \(\frac{8}{11}\): multiply numerator and denominator by 2.
  \[\frac{8}{11} = \frac{8 \times 2}{11 \times 2} = \frac{16}{22}\]
• Step 3. Compare the numerators: 11 vs 16.
• Step 4. Since 11 is less than 16, we know that \(\frac{1}{2} < \frac{8}{11}\).

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: \(1 \times 11 = 11\).
• Step 2. Multiply the numerator of the second fraction by the denominator of the first: \(8 \times 2 = 16\).
• Step 3. Compare the products: \(11 < 16\), so \(\frac{1}{2} < \frac{8}{11}\).

Method 3: Decimal Comparison

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

Since 0.5 is less than 0.727273, we confirm that \(\frac{1}{2} < \frac{8}{11}\). In percentage terms, \(\frac{1}{2}\) is 50% and \(\frac{8}{11}\) is 72.7273%, a difference of 22.7273 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 22, we're cutting both quantities into equal-sized pieces. Then 11 pieces vs 16 pieces is a straightforward comparison.