Is 1/6 Bigger Than 7/9?

Method 1: Common Denominators

To compare these fractions, we need a common denominator. The denominators are 6 and 9, and the least common denominator (LCD) is 18.

• Step 1. Convert \(\frac{1}{6}\): multiply numerator and denominator by 3.
  \[\frac{1}{6} = \frac{1 \times 3}{6 \times 3} = \frac{3}{18}\]
• Step 2. Convert \(\frac{7}{9}\): multiply numerator and denominator by 2.
  \[\frac{7}{9} = \frac{7 \times 2}{9 \times 2} = \frac{14}{18}\]
• Step 3. Compare the numerators: 3 vs 14.
• Step 4. Since 3 is less than 14, we know that \(\frac{1}{6} < \frac{7}{9}\).

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: \(1 \times 9 = 9\).
• Step 2. Multiply the numerator of the second fraction by the denominator of the first: \(7 \times 6 = 42\).
• Step 3. Compare the products: \(9 < 42\), so \(\frac{1}{6} < \frac{7}{9}\).

Method 3: Decimal Comparison

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

Since 0.166667 is less than 0.777778, we confirm that \(\frac{1}{6} < \frac{7}{9}\). In percentage terms, \(\frac{1}{6}\) is 16.6667% and \(\frac{7}{9}\) is 77.7778%, a difference of 61.1111 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 18, we're cutting both quantities into equal-sized pieces. Then 3 pieces vs 14 pieces is a straightforward comparison.