Is 8/9 Bigger Than 9/11?

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 \(\frac{8}{9}\): multiply numerator and denominator by 11.
  \[\frac{8}{9} = \frac{8 \times 11}{9 \times 11} = \frac{88}{99}\]
• Step 2. Convert \(\frac{9}{11}\): multiply numerator and denominator by 9.
  \[\frac{9}{11} = \frac{9 \times 9}{11 \times 9} = \frac{81}{99}\]
• Step 3. Compare the numerators: 88 vs 81.
• Step 4. Since 88 is greater than 81, we know that \(\frac{8}{9} > \frac{9}{11}\).

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: \(8 \times 11 = 88\).
• Step 2. Multiply the numerator of the second fraction by the denominator of the first: \(9 \times 9 = 81\).
• Step 3. Compare the products: \(88 > 81\), so \(\frac{8}{9} > \frac{9}{11}\).

Method 3: Decimal Comparison

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

Since 0.888889 is greater than 0.818182, we confirm that \(\frac{8}{9} > \frac{9}{11}\). In percentage terms, \(\frac{8}{9}\) is 88.8889% and \(\frac{9}{11}\) is 81.8182%, a difference of 7.0707 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 88 pieces vs 81 pieces is a straightforward comparison.