Property of Equivalent Fractions

If both the numerator and the denominator of a fraction are multiplied or divided by the same nonzero number k, the resulting fraction is an equivalent fraction $$ \frac{a}{b} \sim \frac{a \cdot k}{b \cdot k} \sim \frac{a : k}{b : k} \ \ \ \ \ \ with \ \ \ k \ne 0 $$

This property of fractions follows directly from the invariant property of division.

Note. The condition k \ne 0 is required to avoid division by zero, which is undefined. $$ \frac{a \cdot 0}{b \cdot 0} = \frac{0}{0} $$

A practical example

Consider the fraction

$$ \frac{2}{3} $$

If we multiply both the numerator and the denominator by 5, we obtain an equivalent fraction

$$ \frac{2}{3} \sim \frac{2 \cdot 5}{3 \cdot 5} $$

$$ \frac{2}{3} \sim \frac{10}{15} $$

Check. To verify that two fractions are equivalent, we use cross multiplication. $$ \frac{a}{b} \sim \frac{c}{d} \Longleftrightarrow a \cdot d = b \cdot c $$ In this case, $$ \frac{2}{3} \sim \frac{10}{15} \Longleftrightarrow 2 \cdot 15 = 3 \cdot 10 $$ $$ \frac{2}{3} \sim \frac{10}{15} \Longleftrightarrow 30 = 30 $$ The equality holds, so the two fractions are equivalent.

Example 2

Consider the fraction

$$ \frac{12}{8} $$

If we divide both the numerator and the denominator by 4, we obtain an equivalent fraction

$$ \frac{12}{8} \sim \frac{12:4}{8:4} $$

$$ \frac{12}{8} \sim \frac{3}{2} $$

Check. Applying cross multiplication $$ \frac{a}{b} \sim \frac{c}{d} \Longleftrightarrow a \cdot d = b \cdot c $$ In this case, $$ \frac{12}{8} \sim \frac{3}{2} \Longleftrightarrow 12 \cdot 2 = 8 \cdot 3 $$ $$ \frac{12}{8} \sim \frac{3}{2} \Longleftrightarrow 24 = 24 $$ The equality holds, so the two fractions are equivalent.

And so on.