Fractions that name the same rational number are called equivalent fractions, and understanding which fractions equal -2/3 opens the door to deeper number sense and algebraic fluency. In mathematics, two fractions are equivalent when they represent the same point on the number line, even if their numerators and denominators differ. The fraction -2/3 is a negative rational number, and infinitely many other fractions can express its exact value. By multiplying or dividing both the numerator and denominator by the same nonzero integer, while preserving the sign, we generate an endless family of equivalent fractions. This concept is not only foundational for fraction arithmetic but also essential for simplifying expressions, solving equations, and working with ratios in algebra, geometry, and real-world problem solving.
The Definition of Equivalent Fractions Two fractions (\frac{a}{b}) and (\frac{c}{d}) are equivalent if and only if (a \times d = b \times c), cross-multiplying yields the same product. Consider this: applied to -2/3, any fraction (\frac{-2k}{3k}) (where (k) is a nonzero integer) will satisfy this condition. In practice, for example, taking (k = 2) gives (\frac{-4}{6}), and ((-4) \times 3 = -12) while (6 \times (-2) = -12), confirming equivalence. This multiplicative relationship is the engine behind all fraction manipulation and is why recognizing equivalent forms streamlines computation and comparison.
Multiplying by Forms of One A powerful way to generate equivalent fractions is to multiply the given fraction by a form of one, such as (\frac{k}{k}). Since (\frac{k}{k} = 1) for any nonzero (k), the value of the fraction does not change. Starting with (-\frac{2}{3}), multiplying by (\frac{5}{5}) yields (-\frac{10}{15}); multiplying by (\frac{7}{7}) yields (-\frac{14}{21}); multiplying by (\frac{-1}{-1}) yields (\frac{2}{
