Is 2/3 Bigger Than 10/11?

Method 1: Common Denominators

To compare these fractions, we need a common denominator. The denominators are 3 and 11, and the least common denominator (LCD) is 33.

• Step 1. Convert \(\frac{2}{3}\): multiply numerator and denominator by 11.
  \[\frac{2}{3} = \frac{2 \times 11}{3 \times 11} = \frac{22}{33}\]
• Step 2. Convert \(\frac{10}{11}\): multiply numerator and denominator by 3.
  \[\frac{10}{11} = \frac{10 \times 3}{11 \times 3} = \frac{30}{33}\]
• Step 3. Compare the numerators: 22 vs 30.
• Step 4. Since 22 is less than 30, we know that \(\frac{2}{3} < \frac{10}{11}\).

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: \(2 \times 11 = 22\).
• Step 2. Multiply the numerator of the second fraction by the denominator of the first: \(10 \times 3 = 30\).
• Step 3. Compare the products: \(22 < 30\), so \(\frac{2}{3} < \frac{10}{11}\).

Method 3: Decimal Comparison

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

Since 0.666667 is less than 0.909091, we confirm that \(\frac{2}{3} < \frac{10}{11}\). In percentage terms, \(\frac{2}{3}\) is 66.6667% and \(\frac{10}{11}\) is 90.9091%, a difference of 24.2424 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 33, we're cutting both quantities into equal-sized pieces. Then 22 pieces vs 30 pieces is a straightforward comparison.