Is 1/7 Bigger Than 1/9?

Method 1: Common Denominators

To compare these fractions, we need a common denominator. The denominators are 7 and 9, and the least common denominator (LCD) is 63.

• Step 1. Convert \(\frac{1}{7}\): multiply numerator and denominator by 9.
  \[\frac{1}{7} = \frac{1 \times 9}{7 \times 9} = \frac{9}{63}\]
• Step 2. Convert \(\frac{1}{9}\): multiply numerator and denominator by 7.
  \[\frac{1}{9} = \frac{1 \times 7}{9 \times 7} = \frac{7}{63}\]
• Step 3. Compare the numerators: 9 vs 7.
• Step 4. Since 9 is greater than 7, we know that \(\frac{1}{7} > \frac{1}{9}\).

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: \(1 \times 9 = 9\).
• Step 2. Multiply the numerator of the second fraction by the denominator of the first: \(1 \times 7 = 7\).
• Step 3. Compare the products: \(9 > 7\), so \(\frac{1}{7} > \frac{1}{9}\).

Method 3: Decimal Comparison

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

Since 0.142857 is greater than 0.111111, we confirm that \(\frac{1}{7} > \frac{1}{9}\). In percentage terms, \(\frac{1}{7}\) is 14.2857% and \(\frac{1}{9}\) is 11.1111%, a difference of 3.1746 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 9ths are larger pieces than 7ths, \(\frac{1}{7}\) is the bigger fraction even though both have 1 in the numerator.