Is 3/7 Bigger Than 3/10?

Method 1: Common Denominators

To compare these fractions, we need a common denominator. The denominators are 7 and 10, and the least common denominator (LCD) is 70.

• Step 1. Convert \(\frac{3}{7}\): multiply numerator and denominator by 10.
  \[\frac{3}{7} = \frac{3 \times 10}{7 \times 10} = \frac{30}{70}\]
• Step 2. Convert \(\frac{3}{10}\): multiply numerator and denominator by 7.
  \[\frac{3}{10} = \frac{3 \times 7}{10 \times 7} = \frac{21}{70}\]
• Step 3. Compare the numerators: 30 vs 21.
• Step 4. Since 30 is greater than 21, we know that \(\frac{3}{7} > \frac{3}{10}\).

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: \(3 \times 10 = 30\).
• Step 2. Multiply the numerator of the second fraction by the denominator of the first: \(3 \times 7 = 21\).
• Step 3. Compare the products: \(30 > 21\), so \(\frac{3}{7} > \frac{3}{10}\).

Method 3: Decimal Comparison

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

Since 0.428571 is greater than 0.3, we confirm that \(\frac{3}{7} > \frac{3}{10}\). In percentage terms, \(\frac{3}{7}\) is 42.8571% and \(\frac{3}{10}\) is 30%, a difference of 12.8571 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 7ths, \(\frac{3}{7}\) is the bigger fraction even though both have 3 in the numerator.