Is 9/10 Bigger Than 10/11?

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 \(\frac{10}{11}\): multiply numerator and denominator by 10.
  \[\frac{10}{11} = \frac{10 \times 10}{11 \times 10} = \frac{100}{110}\]
• Step 3. Compare the numerators: 99 vs 100.
• Step 4. Since 99 is less than 100, we know that \(\frac{9}{10} < \frac{10}{11}\).

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: \(10 \times 10 = 100\).
• Step 3. Compare the products: \(99 < 100\), so \(\frac{9}{10} < \frac{10}{11}\).

Method 3: Decimal Comparison

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

Since 0.9 is less than 0.909091, we confirm that \(\frac{9}{10} < \frac{10}{11}\). In percentage terms, \(\frac{9}{10}\) is 90% and \(\frac{10}{11}\) is 90.9091%, a difference of 0.9091 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 110, we're cutting both quantities into equal-sized pieces. Then 99 pieces vs 100 pieces is a straightforward comparison.