Is 2/3 Bigger Than 6/7?

Method 1: Common Denominators

To compare these fractions, we need a common denominator. The denominators are 3 and 7, and the least common denominator (LCD) is 21.

• Step 1. Convert \(\frac{2}{3}\): multiply numerator and denominator by 7.
  \[\frac{2}{3} = \frac{2 \times 7}{3 \times 7} = \frac{14}{21}\]
• Step 2. Convert \(\frac{6}{7}\): multiply numerator and denominator by 3.
  \[\frac{6}{7} = \frac{6 \times 3}{7 \times 3} = \frac{18}{21}\]
• Step 3. Compare the numerators: 14 vs 18.
• Step 4. Since 14 is less than 18, we know that \(\frac{2}{3} < \frac{6}{7}\).

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: \(2 \times 7 = 14\).
• Step 2. Multiply the numerator of the second fraction by the denominator of the first: \(6 \times 3 = 18\).
• Step 3. Compare the products: \(14 < 18\), so \(\frac{2}{3} < \frac{6}{7}\).

Method 3: Decimal Comparison

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

Since 0.666667 is less than 0.857143, we confirm that \(\frac{2}{3} < \frac{6}{7}\). In percentage terms, \(\frac{2}{3}\) is 66.6667% and \(\frac{6}{7}\) is 85.7143%, a difference of 19.0476 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 21, we're cutting both quantities into equal-sized pieces. Then 14 pieces vs 18 pieces is a straightforward comparison.