Fractions - Equivalent Fractions

Equivalent fractions are fractions that represent the same value or part of a whole, even though they have different numerators and denominators. For example, if you divide a rectangular bar into 2 equal parts and shade 1, or divide it into 4 equal parts and shade 2, the shaded area remains the same: $\frac{1}{2} = \frac{2}{4}$.

To create an equivalent fraction, you can multiply both the numerator and the denominator by the same non-zero number. Visually, this is like taking an existing diagram and drawing more lines to create smaller sub-sections without changing the total colored portion.

Equivalent fractions can also be found by dividing both the numerator and the denominator by a common factor. This process is known as 'simplification' or 'reducing the fraction'. For instance, if you have $\frac{4}{8}$ of a circle shaded, you can group the 4 parts together to see it is equivalent to $\frac{1}{2}$.

A fraction is in its simplest form (lowest terms) when the numerator and the denominator have no common factors other than 1. On a diagram, the simplest form uses the largest possible equal pieces to represent the portion.

On a number line, all equivalent fractions are represented by the same point. If you mark $\frac{1}{3}$, $\frac{2}{6}$, and $\frac{3}{9}$ on a number line between 0 and 1, all three marks will land on the exact same spot.

The Cross-Multiplication Method is a quick way to check for equivalence. If you have two fractions $\frac{a}{b}$ and $\frac{c}{d}$, they are equivalent if the product of the first numerator and second denominator ($a \times d$) equals the product of the first denominator and second numerator ($b \times c$).

When generating a series of equivalent fractions, you are essentially multiplying the original fraction by $\frac{n}{n}$ (which is equal to 1). Since multiplying any number by 1 does not change its value, the fractions remain equivalent: $\frac{2}{3} \times \frac{2}{2} = \frac{4}{6}$.