Is 7/9 Bigger Than 6/11?

Method 1: Common Denominators

To compare these fractions, we need a common denominator. The denominators are 9 and 11, and the least common denominator (LCD) is 99.

• Step 1. Convert \(\frac{7}{9}\): multiply numerator and denominator by 11.
  \[\frac{7}{9} = \frac{7 \times 11}{9 \times 11} = \frac{77}{99}\]
• Step 2. Convert \(\frac{6}{11}\): multiply numerator and denominator by 9.
  \[\frac{6}{11} = \frac{6 \times 9}{11 \times 9} = \frac{54}{99}\]
• Step 3. Compare the numerators: 77 vs 54.
• Step 4. Since 77 is greater than 54, we know that \(\frac{7}{9} > \frac{6}{11}\).

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: \(7 \times 11 = 77\).
• Step 2. Multiply the numerator of the second fraction by the denominator of the first: \(6 \times 9 = 54\).
• Step 3. Compare the products: \(77 > 54\), so \(\frac{7}{9} > \frac{6}{11}\).

Method 3: Decimal Comparison

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

Since 0.777778 is greater than 0.545455, we confirm that \(\frac{7}{9} > \frac{6}{11}\). In percentage terms, \(\frac{7}{9}\) is 77.7778% and \(\frac{6}{11}\) is 54.5455%, a difference of 23.2323 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 99, we're cutting both quantities into equal-sized pieces. Then 77 pieces vs 54 pieces is a straightforward comparison.