Is 1/9 Bigger Than 9/10?

Method 1: Common Denominators

To compare these fractions, we need a common denominator. The denominators are 9 and 10, and the least common denominator (LCD) is 90.

• Step 1. Convert \(\frac{1}{9}\): multiply numerator and denominator by 10.
  \[\frac{1}{9} = \frac{1 \times 10}{9 \times 10} = \frac{10}{90}\]
• Step 2. Convert \(\frac{9}{10}\): multiply numerator and denominator by 9.
  \[\frac{9}{10} = \frac{9 \times 9}{10 \times 9} = \frac{81}{90}\]
• Step 3. Compare the numerators: 10 vs 81.
• Step 4. Since 10 is less than 81, we know that \(\frac{1}{9} < \frac{9}{10}\).

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: \(1 \times 10 = 10\).
• Step 2. Multiply the numerator of the second fraction by the denominator of the first: \(9 \times 9 = 81\).
• Step 3. Compare the products: \(10 < 81\), so \(\frac{1}{9} < \frac{9}{10}\).

Method 3: Decimal Comparison

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

Since 0.111111 is less than 0.9, we confirm that \(\frac{1}{9} < \frac{9}{10}\). In percentage terms, \(\frac{1}{9}\) is 11.1111% and \(\frac{9}{10}\) is 90%, a difference of 78.8889 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 90, we're cutting both quantities into equal-sized pieces. Then 10 pieces vs 81 pieces is a straightforward comparison.