Is 1/3 Bigger Than 1/10?

Method 1: Common Denominators

To compare these fractions, we need a common denominator. The denominators are 3 and 10, and the least common denominator (LCD) is 30.

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

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: \(1 \times 10 = 10\).
• Step 2. Multiply the numerator of the second fraction by the denominator of the first: \(1 \times 3 = 3\).
• Step 3. Compare the products: \(10 > 3\), so \(\frac{1}{3} > \frac{1}{10}\).

Method 3: Decimal Comparison

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

Since 0.333333 is greater than 0.1, we confirm that \(\frac{1}{3} > \frac{1}{10}\). In percentage terms, \(\frac{1}{3}\) is 33.3333% and \(\frac{1}{10}\) is 10%, a difference of 23.3333 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 10ths are larger pieces than 3ths, \(\frac{1}{3}\) is the bigger fraction even though both have 1 in the numerator.