Is 2/7 Bigger Than 8/11?

Method 1: Common Denominators

To compare these fractions, we need a common denominator. The denominators are 7 and 11, and the least common denominator (LCD) is 77.

• Step 1. Convert \(\frac{2}{7}\): multiply numerator and denominator by 11.
  \[\frac{2}{7} = \frac{2 \times 11}{7 \times 11} = \frac{22}{77}\]
• Step 2. Convert \(\frac{8}{11}\): multiply numerator and denominator by 7.
  \[\frac{8}{11} = \frac{8 \times 7}{11 \times 7} = \frac{56}{77}\]
• Step 3. Compare the numerators: 22 vs 56.
• Step 4. Since 22 is less than 56, we know that \(\frac{2}{7} < \frac{8}{11}\).

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: \(8 \times 7 = 56\).
• Step 3. Compare the products: \(22 < 56\), so \(\frac{2}{7} < \frac{8}{11}\).

Method 3: Decimal Comparison

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

Since 0.285714 is less than 0.727273, we confirm that \(\frac{2}{7} < \frac{8}{11}\). In percentage terms, \(\frac{2}{7}\) is 28.5714% and \(\frac{8}{11}\) is 72.7273%, a difference of 44.1558 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 77, we're cutting both quantities into equal-sized pieces. Then 22 pieces vs 56 pieces is a straightforward comparison.