Is 1/6 Bigger Than 9/10?

Method 1: Common Denominators

To compare these fractions, we need a common denominator. The denominators are 6 and 10, and the least common denominator (LCD) is 30.

• Step 1. Convert \(\frac{1}{6}\): multiply numerator and denominator by 5.
  \[\frac{1}{6} = \frac{1 \times 5}{6 \times 5} = \frac{5}{30}\]
• Step 2. Convert \(\frac{9}{10}\): multiply numerator and denominator by 3.
  \[\frac{9}{10} = \frac{9 \times 3}{10 \times 3} = \frac{27}{30}\]
• Step 3. Compare the numerators: 5 vs 27.
• Step 4. Since 5 is less than 27, we know that \(\frac{1}{6} < \frac{9}{10}\).

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: \(9 \times 6 = 54\).
• Step 3. Compare the products: \(10 < 54\), so \(\frac{1}{6} < \frac{9}{10}\).

Method 3: Decimal Comparison

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

Since 0.166667 is less than 0.9, we confirm that \(\frac{1}{6} < \frac{9}{10}\). In percentage terms, \(\frac{1}{6}\) is 16.6667% and \(\frac{9}{10}\) is 90%, a difference of 73.3333 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 30, we're cutting both quantities into equal-sized pieces. Then 5 pieces vs 27 pieces is a straightforward comparison.