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Imagine your friend offers you 34\frac{3}{4} of a giant chocolate bar, or 56\frac{5}{6} of the exact same bar. Which one gives you more chocolate?
When fractions have different bottom numbers, called denominators, their slices are different sizes — so we need clever strategies to compare them.

The Golden Rule: Same Sized Whole

Before comparing any two fractions, both fractions must refer to the same whole.
Half of a tiny cupcake is smaller than a quarter of a huge cake. We can only compare fractions when the starting wholes are identical!
What is the quickest way to compare two fractions without doing lots of calculations?

Strategy 1: The Halfway Benchmark

With the Benchmark Strategy, fractions are compared by determining their position relative to reference points such as 00, 12\frac{1}{2}, and 11.
Let's compare 38\frac{3}{8} and 46\frac{4}{6} by seeing how each compares to 12\frac{1}{2}.
Since 38<12\frac{3}{8} < \frac{1}{2} and 46>12\frac{4}{6} > \frac{1}{2}, we know 38<46\frac{3}{8} < \frac{4}{6}. Whenever fraction A<12A < \frac{1}{2} and fraction B>12B > \frac{1}{2}, then A<BA < B.
What if both fractions are on the same side of 12\frac{1}{2}? Let's look at a way to make the pieces match.