Is 6/7 Bigger Than 7/12?

Method 1: Common Denominators

To compare these fractions, we need a common denominator. The denominators are 7 and 12, and the least common denominator (LCD) is 84.

• Step 1. Convert \(\frac{6}{7}\): multiply numerator and denominator by 12.
  \[\frac{6}{7} = \frac{6 \times 12}{7 \times 12} = \frac{72}{84}\]
• Step 2. Convert \(\frac{7}{12}\): multiply numerator and denominator by 7.
  \[\frac{7}{12} = \frac{7 \times 7}{12 \times 7} = \frac{49}{84}\]
• Step 3. Compare the numerators: 72 vs 49.
• Step 4. Since 72 is greater than 49, we know that \(\frac{6}{7} > \frac{7}{12}\).

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: \(6 \times 12 = 72\).
• Step 2. Multiply the numerator of the second fraction by the denominator of the first: \(7 \times 7 = 49\).
• Step 3. Compare the products: \(72 > 49\), so \(\frac{6}{7} > \frac{7}{12}\).

Method 3: Decimal Comparison

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

Since 0.857143 is greater than 0.583333, we confirm that \(\frac{6}{7} > \frac{7}{12}\). In percentage terms, \(\frac{6}{7}\) is 85.7143% and \(\frac{7}{12}\) is 58.3333%, a difference of 27.381 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 84, we're cutting both quantities into equal-sized pieces. Then 72 pieces vs 49 pieces is a straightforward comparison.