Two fractions can represent the same value (or be equivalent) if they simplify to the same number when expressed in their lowest terms. This can occur even when the numerators and denominators of the fractions are different.
The concept is based on the idea of equivalent fractions, which states that if you multiply or divide both the numerator and denominator of a fraction by the same non-zero number, you will get an equivalent fraction.
For example, consider the fractions \(\frac{1}{2}\) and \(\frac{2}{4}\). These fractions have different numerators (1 and 2) and denominators (2 and 4), but they describe the same value:
\[ \frac{2}{4} = \frac{2 \div 2}{4 \div 2} = \frac{1}{2} \]
Both fractions represent the same part of a whole, which is one half.
In general, if you have two fractions \(\frac{a}{b}\) and \(\frac{c}{d}\), they are equivalent if:
\[ a \cdot d = b \cdot c \]
This means that the cross-products of the two fractions are equal. For example, for \(\frac{1}{2}\) and \(\frac{2}{4}\):
\[ 1 \cdot 4 = 2 \cdot 2 \implies 4 = 4 \]
Thus, \(\frac{1}{2}\) and \(\frac{2}{4}\) represent the same value, even though their numerators and denominators differ.