Is 1/6 Bigger Than 5/7?

Method 1: Common Denominators

To compare these fractions, we need a common denominator. The denominators are 6 and 7, and the least common denominator (LCD) is 42.

• Step 1. Convert \(\frac{1}{6}\): multiply numerator and denominator by 7.
  \[\frac{1}{6} = \frac{1 \times 7}{6 \times 7} = \frac{7}{42}\]
• Step 2. Convert \(\frac{5}{7}\): multiply numerator and denominator by 6.
  \[\frac{5}{7} = \frac{5 \times 6}{7 \times 6} = \frac{30}{42}\]
• Step 3. Compare the numerators: 7 vs 30.
• Step 4. Since 7 is less than 30, we know that \(\frac{1}{6} < \frac{5}{7}\).

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: \(1 \times 7 = 7\).
• Step 2. Multiply the numerator of the second fraction by the denominator of the first: \(5 \times 6 = 30\).
• Step 3. Compare the products: \(7 < 30\), so \(\frac{1}{6} < \frac{5}{7}\).

Method 3: Decimal Comparison

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

Since 0.166667 is less than 0.714286, we confirm that \(\frac{1}{6} < \frac{5}{7}\). In percentage terms, \(\frac{1}{6}\) is 16.6667% and \(\frac{5}{7}\) is 71.4286%, a difference of 54.7619 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 42, we're cutting both quantities into equal-sized pieces. Then 7 pieces vs 30 pieces is a straightforward comparison.