Is 2/3 Bigger Than 2/7?

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

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: \(2 \times 3 = 6\).
• Step 3. Compare the products: \(14 > 6\), so \(\frac{2}{3} > \frac{2}{7}\).

Method 3: Decimal Comparison

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

Since 0.666667 is greater than 0.285714, we confirm that \(\frac{2}{3} > \frac{2}{7}\). In percentage terms, \(\frac{2}{3}\) is 66.6667% and \(\frac{2}{7}\) is 28.5714%, a difference of 38.0952 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 7ths are larger pieces than 3ths, \(\frac{2}{3}\) is the bigger fraction even though both have 2 in the numerator.