Comparing and Ordering Fractions

When working with fractions, you often need to figure out which one is larger or put a whole group of them in order. There are two great strategies for doing this: using benchmark fractions and finding a common denominator.

Using Benchmark Fractions

A benchmark fraction is a common, familiar fraction like $\frac{1}{2}$ that you can use to quickly estimate and compare other fractions.

Example: Is $\frac{4}{7}$ greater or less than $\frac{1}{2}$?

1. Think about what exactly half of the denominator $7$ is. Half of $7$ is $3.5$.
2. Because the numerator $4$ is larger than $3.5$, the fraction $\frac{4}{7}$ represents more than a half.
3. Answer: $\frac{4}{7} > \frac{1}{2}$

Finding a Common Denominator

When fractions are close in size or hard to estimate, the most accurate method is to give them a common denominator. This means rewriting the fractions so their bottom numbers are exactly the same.

Example: Compare $\frac{3}{8}$ and $\frac{5}{12}$.

1. Find a common multiple for the denominators $8$ and $12$. The number $24$ works perfectly because both $8$ and $12$ divide into it evenly.
2. Convert both fractions to have $24$ as the denominator: $\frac{3}{8} = \frac{3 \times 3}{8 \times 3} = \frac{9}{24}$ $\frac{5}{12} = \frac{5 \times 2}{12 \times 2} = \frac{10}{24}$
3. Now that the pieces are the same size, just compare the numerators. Since $9 < 10$, we know $\frac{9}{24} < \frac{10}{24}$.
4. Answer: $\frac{3}{8} < \frac{5}{12}$

Ordering Fractions

You can use the common denominator method to order a list of fractions from least to greatest (or greatest to least).

Example: Order $\frac{2}{3}$, $\frac{5}{8}$, and $\frac{3}{4}$ from least to greatest.

1. Find a common denominator for $3$, $8$, and $4$. The number $24$ is a multiple of all three.
2. Convert each fraction: $\frac{2}{3} = \frac{16}{24}$ $\frac{5}{8} = \frac{15}{24}$ $\frac{3}{4} = \frac{18}{24}$
3. Look at the new numerators: $15$, $16$, and $18$. Order them from smallest to largest: $15 < 16 < 18$.
4. Answer: The correct order from least to greatest is $\frac{5}{8}$, $\frac{2}{3}$, $\frac{3}{4}$.