Is 3/7 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 \(\frac{3}{7}\): multiply numerator and denominator by 11.
  \[\frac{3}{7} = \frac{3 \times 11}{7 \times 11} = \frac{33}{77}\]
• 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: 33 vs 42.
• Step 4. Since 33 is less than 42, we know that \(\frac{3}{7} < \frac{6}{11}\).

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: \(3 \times 11 = 33\).
• Step 2. Multiply the numerator of the second fraction by the denominator of the first: \(6 \times 7 = 42\).
• Step 3. Compare the products: \(33 < 42\), so \(\frac{3}{7} < \frac{6}{11}\).

Method 3: Decimal Comparison

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

Since 0.428571 is less than 0.545455, we confirm that \(\frac{3}{7} < \frac{6}{11}\). In percentage terms, \(\frac{3}{7}\) is 42.8571% and \(\frac{6}{11}\) is 54.5455%, a difference of 11.6883 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 33 pieces vs 42 pieces is a straightforward comparison.