Is 1/9 Bigger Than 1/12?

Method 1: Common Denominators

To compare these fractions, we need a common denominator. The denominators are 9 and 12, and the least common denominator (LCD) is 36.

• Step 1. Convert \(\frac{1}{9}\): multiply numerator and denominator by 4.
  \[\frac{1}{9} = \frac{1 \times 4}{9 \times 4} = \frac{4}{36}\]
• Step 2. Convert \(\frac{1}{12}\): multiply numerator and denominator by 3.
  \[\frac{1}{12} = \frac{1 \times 3}{12 \times 3} = \frac{3}{36}\]
• Step 3. Compare the numerators: 4 vs 3.
• Step 4. Since 4 is greater than 3, we know that \(\frac{1}{9} > \frac{1}{12}\).

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: \(1 \times 12 = 12\).
• Step 2. Multiply the numerator of the second fraction by the denominator of the first: \(1 \times 9 = 9\).
• Step 3. Compare the products: \(12 > 9\), so \(\frac{1}{9} > \frac{1}{12}\).

Method 3: Decimal Comparison

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

Since 0.111111 is greater than 0.083333, we confirm that \(\frac{1}{9} > \frac{1}{12}\). In percentage terms, \(\frac{1}{9}\) is 11.1111% and \(\frac{1}{12}\) is 8.3333%, a difference of 2.7778 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 12ths are larger pieces than 9ths, \(\frac{1}{9}\) is the bigger fraction even though both have 1 in the numerator.