Is 5/6 Bigger Than 6/7?

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 \(\frac{5}{6}\): multiply numerator and denominator by 7.
  \[\frac{5}{6} = \frac{5 \times 7}{6 \times 7} = \frac{35}{42}\]
• Step 2. Convert \(\frac{6}{7}\): multiply numerator and denominator by 6.
  \[\frac{6}{7} = \frac{6 \times 6}{7 \times 6} = \frac{36}{42}\]
• Step 3. Compare the numerators: 35 vs 36.
• Step 4. Since 35 is less than 36, we know that \(\frac{5}{6} < \frac{6}{7}\).

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: \(5 \times 7 = 35\).
• Step 2. Multiply the numerator of the second fraction by the denominator of the first: \(6 \times 6 = 36\).
• Step 3. Compare the products: \(35 < 36\), so \(\frac{5}{6} < \frac{6}{7}\).

Method 3: Decimal Comparison

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

Since 0.833333 is less than 0.857143, we confirm that \(\frac{5}{6} < \frac{6}{7}\). In percentage terms, \(\frac{5}{6}\) is 83.3333% and \(\frac{6}{7}\) is 85.7143%, a difference of 2.381 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 35 pieces vs 36 pieces is a straightforward comparison.