Is 6/7 Bigger Than 6/11?

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 \(\frac{6}{7}\): multiply numerator and denominator by 11.
  \[\frac{6}{7} = \frac{6 \times 11}{7 \times 11} = \frac{66}{77}\]
• Step 2. Convert \(\frac{6}{11}\): multiply numerator and denominator by 7.
  \[\frac{6}{11} = \frac{6 \times 7}{11 \times 7} = \frac{42}{77}\]
• Step 3. Compare the numerators: 66 vs 42.
• Step 4. Since 66 is greater than 42, we know that \(\frac{6}{7} > \frac{6}{11}\).

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: \(6 \times 11 = 66\).
• Step 2. Multiply the numerator of the second fraction by the denominator of the first: \(6 \times 7 = 42\).
• Step 3. Compare the products: \(66 > 42\), so \(\frac{6}{7} > \frac{6}{11}\).

Method 3: Decimal Comparison

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

Since 0.857143 is greater than 0.545455, we confirm that \(\frac{6}{7} > \frac{6}{11}\). In percentage terms, \(\frac{6}{7}\) is 85.7143% and \(\frac{6}{11}\) is 54.5455%, a difference of 31.1688 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 11ths are larger pieces than 7ths, \(\frac{6}{7}\) is the bigger fraction even though both have 6 in the numerator.