Skill: Use benchmark fractions

A benchmark fraction is a familiar fraction you use to judge the size of another fraction. The most useful benchmarks are $0$, $\frac{1}{2}$, and $1$.

To compare a fraction with $\frac{1}{2}$:

• Find half of the denominator.
• Compare the numerator with that number.
• If the numerator is smaller, the fraction is less than $\frac{1}{2}$.
• If the numerator is larger, the fraction is greater than $\frac{1}{2}$.

Example

Which fraction is greater: $\frac{5}{8}$ or $\frac{3}{7}$?

Step 1: Compare $\frac{5}{8}$ with $\frac{1}{2}$.

Half of $8$ is $4$. Since $5>4$,

$$\frac{5}{8}>\frac{1}{2}.$$

So, $\frac{5}{8}$ is to the right of $\frac{1}{2}$ on a number line.

Step 2: Compare $\frac{3}{7}$ with $\frac{1}{2}$.

Half of $7$ is $3.5$. Since $3<3.5$,

$$\frac{3}{7}<\frac{1}{2}.$$

So, $\frac{3}{7}$ is to the left of $\frac{1}{2}$ on a number line.

Step 3: Compare the fractions using the benchmark.

Since $\frac{5}{8}$ is greater than $\frac{1}{2}$ and $\frac{3}{7}$ is less than $\frac{1}{2}$,

$$\boxed{\frac{5}{8}>\frac{3}{7}}.$$

Using $\frac{1}{2}$ as a benchmark helps you compare fractions without judging them only by their numerators or denominators.