Is 3/7 Bigger Than 7/9?

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 \(\frac{3}{7}\): multiply numerator and denominator by 9.
  \[\frac{3}{7} = \frac{3 \times 9}{7 \times 9} = \frac{27}{63}\]
• Step 2. Convert \(\frac{7}{9}\): multiply numerator and denominator by 7.
  \[\frac{7}{9} = \frac{7 \times 7}{9 \times 7} = \frac{49}{63}\]
• Step 3. Compare the numerators: 27 vs 49.
• Step 4. Since 27 is less than 49, we know that \(\frac{3}{7} < \frac{7}{9}\).

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: \(3 \times 9 = 27\).
• Step 2. Multiply the numerator of the second fraction by the denominator of the first: \(7 \times 7 = 49\).
• Step 3. Compare the products: \(27 < 49\), so \(\frac{3}{7} < \frac{7}{9}\).

Method 3: Decimal Comparison

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

Since 0.428571 is less than 0.777778, we confirm that \(\frac{3}{7} < \frac{7}{9}\). In percentage terms, \(\frac{3}{7}\) is 42.8571% and \(\frac{7}{9}\) is 77.7778%, a difference of 34.9206 percentage points.

Why Does This Work?

These fractions have different numerators and different denominators, so we can't compare them directly. By converting to a common denominator of 63, we're cutting both quantities into equal-sized pieces. Then 27 pieces vs 49 pieces is a straightforward comparison.