Fractions - Equivalent Fractions

Definition of Equivalent Fractions: Equivalent fractions are fractions that represent the same part of a whole or the same value, even though they have different numerators and denominators. Visually, if you have two identical circles, shading $\frac{1}{2}$ of the first circle covers the same area as shading $\frac{2}{4}$ or $\frac{4}{8}$ of the second circle.

Finding Equivalents by Multiplication: You can create an equivalent fraction by multiplying both the numerator and the denominator by the same non-zero number. For example, to find a fraction equivalent to $\frac{1}{3}$, you can multiply both top and bottom by $2$ to get $\frac{2}{6}$. This is like taking a chocolate bar divided into $3$ large pieces and cutting each piece into $2$ smaller pieces to get $6$ total pieces.

Finding Equivalents by Division: If both the numerator and denominator have a common factor, you can divide them by that same number to get an equivalent fraction. This process is often called simplifying or reducing. For instance, $\frac{10}{20}$ can be divided by $10$ on both sides to get $\frac{1}{2}$, showing that $10$ out of $20$ items is the same as half the items.

The Property of One: Any fraction where the numerator and denominator are the same, such as $\frac{2}{2}$, $\frac{5}{5}$, or $\frac{100}{100}$, is equal to $1$. Multiplying a fraction by $\frac{n}{n}$ does not change its value, only its appearance, which is why the resulting fraction is equivalent.

Equivalent Fractions on a Number Line: On a number line marked from $0$ to $1$, equivalent fractions occupy the exact same point. For example, if you divide the line into $4$ parts, the second mark is $\frac{2}{4}$. If you divide the same line into $2$ parts, the first mark is $\frac{1}{2}$. Both marks sit at the exact center of the line.

Checking Equivalence using Cross-Multiplication: To check if two fractions $\frac{a}{b}$ and $\frac{c}{d}$ are equivalent, you can multiply the numerator of the first by the denominator of the second ($a \times d$) and the denominator of the first by the numerator of the second ($b \times c$). If the two products are equal, the fractions are equivalent.

Simplest Form: A fraction is in its simplest form when the only common factor between the numerator and the denominator is $1$. For example, $\frac{3}{4}$ is in simplest form because no number other than $1$ can divide both $3$ and $4$ exactly.