Is 1/2 Bigger Than 9/10?

Method 1: Common Denominators

To compare these fractions, we need a common denominator. The denominators are 2 and 10, and the least common denominator (LCD) is 10.

• Step 1. Convert \(\frac{1}{2}\): multiply numerator and denominator by 5.
  \[\frac{1}{2} = \frac{1 \times 5}{2 \times 5} = \frac{5}{10}\]
• Step 2. Keep \(\frac{9}{10}\) as it is — the denominator is already 10.
• Step 3. Compare the numerators: 5 vs 9.
• Step 4. Since 5 is less than 9, we know that \(\frac{1}{2} < \frac{9}{10}\).

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: \(1 \times 10 = 10\).
• Step 2. Multiply the numerator of the second fraction by the denominator of the first: \(9 \times 2 = 18\).
• Step 3. Compare the products: \(10 < 18\), so \(\frac{1}{2} < \frac{9}{10}\).

Method 3: Decimal Comparison

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

Since 0.5 is less than 0.9, we confirm that \(\frac{1}{2} < \frac{9}{10}\). In percentage terms, \(\frac{1}{2}\) is 50% and \(\frac{9}{10}\) is 90%, a difference of 40 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 10, we're cutting both quantities into equal-sized pieces. Then 5 pieces vs 9 pieces is a straightforward comparison.