Is 9/10 Bigger Than 9/11?

Method 1: Common Denominators

To compare these fractions, we need a common denominator. The denominators are 10 and 11, and the least common denominator (LCD) is 110.

• Step 1. Convert \(\frac{9}{10}\): multiply numerator and denominator by 11.
  \[\frac{9}{10} = \frac{9 \times 11}{10 \times 11} = \frac{99}{110}\]
• Step 2. Convert \(\frac{9}{11}\): multiply numerator and denominator by 10.
  \[\frac{9}{11} = \frac{9 \times 10}{11 \times 10} = \frac{90}{110}\]
• Step 3. Compare the numerators: 99 vs 90.
• Step 4. Since 99 is greater than 90, we know that \(\frac{9}{10} > \frac{9}{11}\).

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: \(9 \times 11 = 99\).
• Step 2. Multiply the numerator of the second fraction by the denominator of the first: \(9 \times 10 = 90\).
• Step 3. Compare the products: \(99 > 90\), so \(\frac{9}{10} > \frac{9}{11}\).

Method 3: Decimal Comparison

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

Since 0.9 is greater than 0.818182, we confirm that \(\frac{9}{10} > \frac{9}{11}\). In percentage terms, \(\frac{9}{10}\) is 90% and \(\frac{9}{11}\) is 81.8182%, a difference of 8.1818 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 11ths are larger pieces than 10ths, \(\frac{9}{10}\) is the bigger fraction even though both have 9 in the numerator.