Is 3/5 Bigger Than 10/11?

Method 1: Common Denominators

To compare these fractions, we need a common denominator. The denominators are 5 and 11, and the least common denominator (LCD) is 55.

• Step 1. Convert \(\frac{3}{5}\): multiply numerator and denominator by 11.
  \[\frac{3}{5} = \frac{3 \times 11}{5 \times 11} = \frac{33}{55}\]
• Step 2. Convert \(\frac{10}{11}\): multiply numerator and denominator by 5.
  \[\frac{10}{11} = \frac{10 \times 5}{11 \times 5} = \frac{50}{55}\]
• Step 3. Compare the numerators: 33 vs 50.
• Step 4. Since 33 is less than 50, we know that \(\frac{3}{5} < \frac{10}{11}\).

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: \(3 \times 11 = 33\).
• Step 2. Multiply the numerator of the second fraction by the denominator of the first: \(10 \times 5 = 50\).
• Step 3. Compare the products: \(33 < 50\), so \(\frac{3}{5} < \frac{10}{11}\).

Method 3: Decimal Comparison

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

Since 0.6 is less than 0.909091, we confirm that \(\frac{3}{5} < \frac{10}{11}\). In percentage terms, \(\frac{3}{5}\) is 60% and \(\frac{10}{11}\) is 90.9091%, a difference of 30.9091 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 55, we're cutting both quantities into equal-sized pieces. Then 33 pieces vs 50 pieces is a straightforward comparison.