Is 1/3 Bigger Than 9/11?

Method 1: Common Denominators

To compare these fractions, we need a common denominator. The denominators are 3 and 11, and the least common denominator (LCD) is 33.

• Step 1. Convert \(\frac{1}{3}\): multiply numerator and denominator by 11.
  \[\frac{1}{3} = \frac{1 \times 11}{3 \times 11} = \frac{11}{33}\]
• Step 2. Convert \(\frac{9}{11}\): multiply numerator and denominator by 3.
  \[\frac{9}{11} = \frac{9 \times 3}{11 \times 3} = \frac{27}{33}\]
• Step 3. Compare the numerators: 11 vs 27.
• Step 4. Since 11 is less than 27, we know that \(\frac{1}{3} < \frac{9}{11}\).

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: \(9 \times 3 = 27\).
• Step 3. Compare the products: \(11 < 27\), so \(\frac{1}{3} < \frac{9}{11}\).

Method 3: Decimal Comparison

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

Since 0.333333 is less than 0.818182, we confirm that \(\frac{1}{3} < \frac{9}{11}\). In percentage terms, \(\frac{1}{3}\) is 33.3333% and \(\frac{9}{11}\) is 81.8182%, a difference of 48.4848 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 33, we're cutting both quantities into equal-sized pieces. Then 11 pieces vs 27 pieces is a straightforward comparison.