Is 1/7 Bigger Than 11/12?

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 \(\frac{1}{7}\): multiply numerator and denominator by 12.
  \[\frac{1}{7} = \frac{1 \times 12}{7 \times 12} = \frac{12}{84}\]
• Step 2. Convert \(\frac{11}{12}\): multiply numerator and denominator by 7.
  \[\frac{11}{12} = \frac{11 \times 7}{12 \times 7} = \frac{77}{84}\]
• Step 3. Compare the numerators: 12 vs 77.
• Step 4. Since 12 is less than 77, we know that \(\frac{1}{7} < \frac{11}{12}\).

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: \(1 \times 12 = 12\).
• Step 2. Multiply the numerator of the second fraction by the denominator of the first: \(11 \times 7 = 77\).
• Step 3. Compare the products: \(12 < 77\), so \(\frac{1}{7} < \frac{11}{12}\).

Method 3: Decimal Comparison

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

Since 0.142857 is less than 0.916667, we confirm that \(\frac{1}{7} < \frac{11}{12}\). In percentage terms, \(\frac{1}{7}\) is 14.2857% and \(\frac{11}{12}\) is 91.6667%, a difference of 77.381 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 12 pieces vs 77 pieces is a straightforward comparison.