Is 6/7 Bigger Than 5/8?

Method 1: Common Denominators

To compare these fractions, we need a common denominator. The denominators are 7 and 8, and the least common denominator (LCD) is 56.

• Step 1. Convert \(\frac{6}{7}\): multiply numerator and denominator by 8.
  \[\frac{6}{7} = \frac{6 \times 8}{7 \times 8} = \frac{48}{56}\]
• Step 2. Convert \(\frac{5}{8}\): multiply numerator and denominator by 7.
  \[\frac{5}{8} = \frac{5 \times 7}{8 \times 7} = \frac{35}{56}\]
• Step 3. Compare the numerators: 48 vs 35.
• Step 4. Since 48 is greater than 35, we know that \(\frac{6}{7} > \frac{5}{8}\).

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: \(6 \times 8 = 48\).
• Step 2. Multiply the numerator of the second fraction by the denominator of the first: \(5 \times 7 = 35\).
• Step 3. Compare the products: \(48 > 35\), so \(\frac{6}{7} > \frac{5}{8}\).

Method 3: Decimal Comparison

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

Since 0.857143 is greater than 0.625, we confirm that \(\frac{6}{7} > \frac{5}{8}\). In percentage terms, \(\frac{6}{7}\) is 85.7143% and \(\frac{5}{8}\) is 62.5%, a difference of 23.2143 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 56, we're cutting both quantities into equal-sized pieces. Then 48 pieces vs 35 pieces is a straightforward comparison.