Is 1/9 Bigger Than 7/12?

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 \(\frac{7}{12}\): multiply numerator and denominator by 3.
  \[\frac{7}{12} = \frac{7 \times 3}{12 \times 3} = \frac{21}{36}\]
• Step 3. Compare the numerators: 4 vs 21.
• Step 4. Since 4 is less than 21, we know that \(\frac{1}{9} < \frac{7}{12}\).

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: \(7 \times 9 = 63\).
• Step 3. Compare the products: \(12 < 63\), so \(\frac{1}{9} < \frac{7}{12}\).

Method 3: Decimal Comparison

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

Since 0.111111 is less than 0.583333, we confirm that \(\frac{1}{9} < \frac{7}{12}\). In percentage terms, \(\frac{1}{9}\) is 11.1111% and \(\frac{7}{12}\) is 58.3333%, a difference of 47.2222 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 36, we're cutting both quantities into equal-sized pieces. Then 4 pieces vs 21 pieces is a straightforward comparison.