Is 4/7 Bigger Than 4/9?

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 \(\frac{4}{7}\): multiply numerator and denominator by 9.
  \[\frac{4}{7} = \frac{4 \times 9}{7 \times 9} = \frac{36}{63}\]
• Step 2. Convert \(\frac{4}{9}\): multiply numerator and denominator by 7.
  \[\frac{4}{9} = \frac{4 \times 7}{9 \times 7} = \frac{28}{63}\]
• Step 3. Compare the numerators: 36 vs 28.
• Step 4. Since 36 is greater than 28, we know that \(\frac{4}{7} > \frac{4}{9}\).

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: \(4 \times 9 = 36\).
• Step 2. Multiply the numerator of the second fraction by the denominator of the first: \(4 \times 7 = 28\).
• Step 3. Compare the products: \(36 > 28\), so \(\frac{4}{7} > \frac{4}{9}\).

Method 3: Decimal Comparison

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

Since 0.571429 is greater than 0.444444, we confirm that \(\frac{4}{7} > \frac{4}{9}\). In percentage terms, \(\frac{4}{7}\) is 57.1429% and \(\frac{4}{9}\) is 44.4444%, a difference of 12.6984 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, \(\frac{4}{7}\) is the bigger fraction even though both have 4 in the numerator.