Is 1/6 Bigger Than 9/11?

Method 1: Common Denominators

To compare these fractions, we need a common denominator. The denominators are 6 and 11, and the least common denominator (LCD) is 66.

• Step 1. Convert \(\frac{1}{6}\): multiply numerator and denominator by 11.
  \[\frac{1}{6} = \frac{1 \times 11}{6 \times 11} = \frac{11}{66}\]
• Step 2. Convert \(\frac{9}{11}\): multiply numerator and denominator by 6.
  \[\frac{9}{11} = \frac{9 \times 6}{11 \times 6} = \frac{54}{66}\]
• Step 3. Compare the numerators: 11 vs 54.
• Step 4. Since 11 is less than 54, we know that \(\frac{1}{6} < \frac{9}{11}\).

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: \(9 \times 6 = 54\).
• Step 3. Compare the products: \(11 < 54\), so \(\frac{1}{6} < \frac{9}{11}\).

Method 3: Decimal Comparison

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

Since 0.166667 is less than 0.818182, we confirm that \(\frac{1}{6} < \frac{9}{11}\). In percentage terms, \(\frac{1}{6}\) is 16.6667% and \(\frac{9}{11}\) is 81.8182%, a difference of 65.1515 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 66, we're cutting both quantities into equal-sized pieces. Then 11 pieces vs 54 pieces is a straightforward comparison.