Is 7/9 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 \(\frac{7}{9}\): multiply numerator and denominator by 4.
  \[\frac{7}{9} = \frac{7 \times 4}{9 \times 4} = \frac{28}{36}\]
• 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: 28 vs 21.
• Step 4. Since 28 is greater than 21, we know that \(\frac{7}{9} > \frac{7}{12}\).

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: \(7 \times 12 = 84\).
• Step 2. Multiply the numerator of the second fraction by the denominator of the first: \(7 \times 9 = 63\).
• Step 3. Compare the products: \(84 > 63\), so \(\frac{7}{9} > \frac{7}{12}\).

Method 3: Decimal Comparison

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

Since 0.777778 is greater than 0.583333, we confirm that \(\frac{7}{9} > \frac{7}{12}\). In percentage terms, \(\frac{7}{9}\) is 77.7778% and \(\frac{7}{12}\) is 58.3333%, a difference of 19.4444 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, \(\frac{7}{9}\) is the bigger fraction even though both have 7 in the numerator.