Is 3/10 Bigger Than 3/11?

Method 1: Common Denominators

To compare these fractions, we need a common denominator. The denominators are 10 and 11, and the least common denominator (LCD) is 110.

• Step 1. Convert \(\frac{3}{10}\): multiply numerator and denominator by 11.
  \[\frac{3}{10} = \frac{3 \times 11}{10 \times 11} = \frac{33}{110}\]
• Step 2. Convert \(\frac{3}{11}\): multiply numerator and denominator by 10.
  \[\frac{3}{11} = \frac{3 \times 10}{11 \times 10} = \frac{30}{110}\]
• Step 3. Compare the numerators: 33 vs 30.
• Step 4. Since 33 is greater than 30, we know that \(\frac{3}{10} > \frac{3}{11}\).

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: \(3 \times 10 = 30\).
• Step 3. Compare the products: \(33 > 30\), so \(\frac{3}{10} > \frac{3}{11}\).

Method 3: Decimal Comparison

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

Since 0.3 is greater than 0.272727, we confirm that \(\frac{3}{10} > \frac{3}{11}\). In percentage terms, \(\frac{3}{10}\) is 30% and \(\frac{3}{11}\) is 27.2727%, a difference of 2.7273 percentage points.

Why Does This Work?

When two fractions have the same numerator, you have the same number of pieces — but the pieces are different sizes. A smaller denominator means each piece is larger. Since 11ths are larger pieces than 10ths, \(\frac{3}{10}\) is the bigger fraction even though both have 3 in the numerator.