Fractions and Decimals - Equivalent Fractions and Simplification

Equivalent Fractions are different fractions that represent the same value or part of a whole. Visually, if you have a circular pizza cut into 2 equal slices and you eat 1, you eat the same amount as if the pizza were cut into 4 equal slices and you ate 2. This is written as $\frac{1}{2} = \frac{2}{4}$.

To find an equivalent fraction, you must multiply or divide both the numerator (top number) and the denominator (bottom number) by the same non-zero number. This is based on the Multiplicative Identity Property, because multiplying by $\frac{n}{n}$ is the same as multiplying by 1.

Simplifying a fraction means reducing it to its 'lowest terms' where the numerator and denominator are as small as possible. Visualise a bar model of $\frac{6}{9}$ where 6 small blocks are shaded out of 9. If you group the blocks into sets of 3, you see that 2 larger blocks are shaded out of 3, resulting in $\frac{2}{3}$.

The Greatest Common Factor (GCF) is the largest number that divides exactly into both the numerator and the denominator. Dividing both parts of the fraction by the GCF is the most efficient way to simplify a fraction in one single step.

A fraction is in its simplest form when the only common factor between the numerator and the denominator is 1. For example, $\frac{5}{7}$ is in simplest form because no whole number other than 1 can divide both 5 and 7 evenly.

On a number line, equivalent fractions occupy the exact same position. If you draw a number line from 0 to 1, the point for $\frac{1}{2}$ will be in the exact same spot as the points for $\frac{2}{4}$, $\frac{3}{6}$, and $\frac{50}{100}$.

Cross-multiplication is a method used to check if two fractions are equivalent. If you have two fractions $\frac{a}{b}$ and $\frac{c}{d}$, they are equivalent if the product of the 'top-left' and 'bottom-right' equals the product of the 'bottom-left' and 'top-right' ($a \times d = b \times c$).