Is 10/11 Bigger Than 7/12?

Method 1: Common Denominators

To compare these fractions, we need a common denominator. The denominators are 11 and 12, and the least common denominator (LCD) is 132.

• Step 1. Convert \(\frac{10}{11}\): multiply numerator and denominator by 12.
  \[\frac{10}{11} = \frac{10 \times 12}{11 \times 12} = \frac{120}{132}\]
• Step 2. Convert \(\frac{7}{12}\): multiply numerator and denominator by 11.
  \[\frac{7}{12} = \frac{7 \times 11}{12 \times 11} = \frac{77}{132}\]
• Step 3. Compare the numerators: 120 vs 77.
• Step 4. Since 120 is greater than 77, we know that \(\frac{10}{11} > \frac{7}{12}\).

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: \(10 \times 12 = 120\).
• Step 2. Multiply the numerator of the second fraction by the denominator of the first: \(7 \times 11 = 77\).
• Step 3. Compare the products: \(120 > 77\), so \(\frac{10}{11} > \frac{7}{12}\).

Method 3: Decimal Comparison

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

Since 0.909091 is greater than 0.583333, we confirm that \(\frac{10}{11} > \frac{7}{12}\). In percentage terms, \(\frac{10}{11}\) is 90.9091% and \(\frac{7}{12}\) is 58.3333%, a difference of 32.5758 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 132, we're cutting both quantities into equal-sized pieces. Then 120 pieces vs 77 pieces is a straightforward comparison.