Fractions - Comparing and Ordering Fractions

Like Fractions: These are fractions that have the same denominator, such as $\frac{2}{7}$ and $\frac{5}{7}$. To compare them, simply look at the numerators. Since $5 > 2$, $\frac{5}{7}$ is greater than $\frac{2}{7}$. Visually, if you have two identical rectangular bars both divided into 7 equal parts, shading 5 parts covers more area than shading 2 parts.

Unlike Fractions with Same Numerators: When two fractions have the same numerator but different denominators, the fraction with the smaller denominator is the larger fraction. For example, $\frac{1}{3} > \frac{1}{5}$. Imagine two pizzas of the same size: if you cut one into 3 slices and another into 5, the slices from the pizza cut into 3 will be much larger.

Unit Fractions: These are fractions where the numerator is always $1$. Examples include $\frac{1}{2}, \frac{1}{4}, \frac{1}{10}$. As the denominator increases, the value of the unit fraction decreases. On a number line between $0$ and $1$, $\frac{1}{2}$ is exactly in the middle, while $\frac{1}{10}$ is much closer to $0$.

Comparing Unlike Fractions: To compare fractions with different numerators and denominators, such as $\frac{2}{3}$ and $\frac{3}{4}$, we first find a common denominator (usually the Least Common Multiple). By converting them to equivalent fractions with the same denominator, we can easily see which one is larger.

Ascending Order: This means arranging fractions from the smallest to the largest. It is like climbing up a ladder. For example, the set $\frac{1}{9}, \frac{4}{9}, \frac{7}{9}$ is in ascending order. We use the 'less than' symbol ($<$) to show this relationship: $\frac{1}{9} < \frac{4}{9} < \frac{7}{9}$.

Descending Order: This means arranging fractions from the largest to the smallest. It is like coming down a slide. For example, the set $\frac{5}{6}, \frac{3}{6}, \frac{1}{6}$ is in descending order. We use the 'greater than' symbol ($>$) to show this: $\frac{5}{6} > \frac{3}{6} > \frac{1}{6}$.

Cross-Multiplication Method: For any two fractions $\frac{a}{b}$ and $\frac{c}{d}$, we can compare them by multiplying the numerator of the first by the denominator of the second ($a \times d$) and the numerator of the second by the denominator of the first ($c \times b$). If $a \times d > b \times c$, then $\frac{a}{b} > \frac{c}{d}$.