Is 3/5 Bigger Than 8/9?

Method 1: Common Denominators

To compare these fractions, we need a common denominator. The denominators are 5 and 9, and the least common denominator (LCD) is 45.

• Step 1. Convert \(\frac{3}{5}\): multiply numerator and denominator by 9.
  \[\frac{3}{5} = \frac{3 \times 9}{5 \times 9} = \frac{27}{45}\]
• Step 2. Convert \(\frac{8}{9}\): multiply numerator and denominator by 5.
  \[\frac{8}{9} = \frac{8 \times 5}{9 \times 5} = \frac{40}{45}\]
• Step 3. Compare the numerators: 27 vs 40.
• Step 4. Since 27 is less than 40, we know that \(\frac{3}{5} < \frac{8}{9}\).

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: \(8 \times 5 = 40\).
• Step 3. Compare the products: \(27 < 40\), so \(\frac{3}{5} < \frac{8}{9}\).

Method 3: Decimal Comparison

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

Since 0.6 is less than 0.888889, we confirm that \(\frac{3}{5} < \frac{8}{9}\). In percentage terms, \(\frac{3}{5}\) is 60% and \(\frac{8}{9}\) is 88.8889%, a difference of 28.8889 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 45, we're cutting both quantities into equal-sized pieces. Then 27 pieces vs 40 pieces is a straightforward comparison.