Comparing and Ordering Fractions

Concept Explanation

Comparing fractions means determining which fraction represents a greater or lesser value. There are several strategies for comparing fractions:

1. Using common denominators: Convert fractions to equivalent fractions with the same denominator, then compare the numerators.
2. Using benchmark fractions: Compare fractions to familiar values like 0, 1/2, or 1.
3. Cross multiplication: Multiply the numerator of each fraction by the denominator of the other fraction.

Worked Examples

Example 1

Problem: Compare $\frac{3}{4}$ and $\frac{2}{3}$ using the correct symbol (< > or =).
Solution: $\frac{3}{4} > \frac{2}{3}$
Explanation: Convert to common denominator: $\frac{3}{4} = \frac{9}{12}$ and $\frac{2}{3} = \frac{8}{12}$. Since 9 > 8, $\frac{3}{4} > \frac{2}{3}$.

Example 2

Problem: Order these fractions from least to greatest: $\frac{1}{2}$, $\frac{3}{8}$, $\frac{5}{8}$
Solution: $\frac{3}{8}$, $\frac{1}{2}$, $\frac{5}{8}$
Explanation: Convert to common denominator: $\frac{1}{2} = \frac{4}{8}$. Now compare $\frac{3}{8}$, $\frac{4}{8}$, and $\frac{5}{8}$.

Example 3

Problem: Compare $\frac{4}{5}$ and $\frac{7}{8}$ using the correct symbol.
Solution: $\frac{4}{5} < \frac{7}{8}$
Explanation: Cross multiply: 4 × 8 = 32 and 5 × 7 = 35. Since 32 < 35, $\frac{4}{5} < \frac{7}{8}$.

Common Errors

Practice Problems

1. Problem:

   Compare $\frac{2} {5}$ and $\frac{3} {7}$ using the correct symbol.

   Solution:

   $\frac{2} {5} > \frac{3} {7}$

2. Problem:

   Order from greatest to least: $\frac{5} {6}$, $\frac{7} {12}$, $\frac{2} {3}$

   Solution:

   $\frac{5} {6}$, $\frac{2} {3}$, $\frac{7} {12}$

3. Problem:

   Compare $\frac{4} {9}$ and $\frac{5} {11}$ using the correct symbol.

   Solution:

   $\frac{4} {9} > \frac{5} {11}$

4. Problem:

   Which is greater: $\frac{3} {5}$ or $\frac{7} {10}$?

   Solution:

   $\frac{7} {10}$

5. Problem:

   Order from least to greatest: $\frac{1} {3}$, $\frac{2} {5}$, $\frac{3} {8}$

   Solution:

   $\frac{3} {8}$, $\frac{1} {3}$, $\frac{2} {5}$

Real-World Application Example

Comparing fractions is essential in many real-life situations, such as cooking (determining if you have enough ingredients), sharing (ensuring fair distribution), and construction (measuring and cutting materials). For example, when following a recipe that calls for 3/4 cup of flour but you only have a 2/3 cup measuring cup, knowing which fraction is larger helps you determine whether you need to add more or have enough.