Fractions - Comparing Like and Unlike Fractions

Understanding Fractions: A fraction is a part of a whole, written in the form $\frac{a}{b}$, where $a$ is the numerator (parts taken) and $b$ is the denominator (total parts). Visually, if you divide a rectangular chocolate bar into $4$ equal pieces and eat $1$, the fraction eaten is $\frac{1}{4}$.

Like Fractions: These are fractions that have the same denominator, such as $\frac{2}{7}$ and $\frac{5}{7}$. When denominators are the same, the fraction with the larger numerator is the greater fraction. Visually, if two identical circles are both divided into $7$ equal slices, $5$ slices will cover a larger area than $2$ slices.

Unlike Fractions with Same Numerator: If two fractions have the same numerator but different denominators (e.g., $\frac{1}{3}$ and $\frac{1}{5}$), the fraction with the smaller denominator is larger. Visually, if you cut a pizza into $3$ large slices, one slice is bigger than if you cut the same pizza into $5$ smaller slices.

Unlike Fractions with Different Denominators: To compare fractions where both parts are different (e.g., $\frac{2}{3}$ and $\frac{3}{4}$), we must first make them 'like' fractions by finding a common denominator, usually the Least Common Multiple (LCM) of the denominators.

Equivalent Fractions: These are fractions that represent the same value, even if they look different. For example, $\frac{1}{2}$, $\frac{2}{4}$, and $\frac{4}{8}$ are all equivalent. Visually, shading half of a circle looks the same regardless of whether it is divided into $2$ large parts or $8$ small parts.

The Cross-Multiplication Rule: A quick way to compare any two fractions $\frac{a}{b}$ and $\frac{c}{d}$ is to cross-multiply. We compare the products $a \times d$ and $b \times c$. If $a \times d$ is greater, then $\frac{a}{b}$ is the larger fraction.

Ordering Fractions: Fractions can be arranged in Ascending Order (smallest to largest) or Descending Order (largest to smallest) by converting all given unlike fractions into like fractions using their LCM and then comparing their numerators.