Is 2/5 Bigger Than 5/7?

Method 1: Common Denominators

To compare these fractions, we need a common denominator. The denominators are 5 and 7, and the least common denominator (LCD) is 35.

• Step 1. Convert \(\frac{2}{5}\): multiply numerator and denominator by 7.
  \[\frac{2}{5} = \frac{2 \times 7}{5 \times 7} = \frac{14}{35}\]
• Step 2. Convert \(\frac{5}{7}\): multiply numerator and denominator by 5.
  \[\frac{5}{7} = \frac{5 \times 5}{7 \times 5} = \frac{25}{35}\]
• Step 3. Compare the numerators: 14 vs 25.
• Step 4. Since 14 is less than 25, we know that \(\frac{2}{5} < \frac{5}{7}\).

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: \(2 \times 7 = 14\).
• Step 2. Multiply the numerator of the second fraction by the denominator of the first: \(5 \times 5 = 25\).
• Step 3. Compare the products: \(14 < 25\), so \(\frac{2}{5} < \frac{5}{7}\).

Method 3: Decimal Comparison

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

Since 0.4 is less than 0.714286, we confirm that \(\frac{2}{5} < \frac{5}{7}\). In percentage terms, \(\frac{2}{5}\) is 40% and \(\frac{5}{7}\) is 71.4286%, a difference of 31.4286 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 35, we're cutting both quantities into equal-sized pieces. Then 14 pieces vs 25 pieces is a straightforward comparison.