Is 5/7 Bigger Than 5/9?

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 \(\frac{5}{7}\): multiply numerator and denominator by 9.
  \[\frac{5}{7} = \frac{5 \times 9}{7 \times 9} = \frac{45}{63}\]
• Step 2. Convert \(\frac{5}{9}\): multiply numerator and denominator by 7.
  \[\frac{5}{9} = \frac{5 \times 7}{9 \times 7} = \frac{35}{63}\]
• Step 3. Compare the numerators: 45 vs 35.
• Step 4. Since 45 is greater than 35, we know that \(\frac{5}{7} > \frac{5}{9}\).

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: \(5 \times 9 = 45\).
• Step 2. Multiply the numerator of the second fraction by the denominator of the first: \(5 \times 7 = 35\).
• Step 3. Compare the products: \(45 > 35\), so \(\frac{5}{7} > \frac{5}{9}\).

Method 3: Decimal Comparison

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

Since 0.714286 is greater than 0.555556, we confirm that \(\frac{5}{7} > \frac{5}{9}\). In percentage terms, \(\frac{5}{7}\) is 71.4286% and \(\frac{5}{9}\) is 55.5556%, a difference of 15.873 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, \(\frac{5}{7}\) is the bigger fraction even though both have 5 in the numerator.