Are two fractions equal? The cross-product decides

The rule

$\dfrac{a}{b} = \dfrac{c}{d} \iff a \times d = b \times c$

Snappy example

Are $\dfrac{6}{9}$ and $\dfrac{8}{12}$ equal?

Cross product: $6 \times 12 = 72$ and $9 \times 8 = 72$. Equal → yes, the fractions are equal (both are $2/3$).

Why it works

Multiply both sides of $\dfrac{a}{b} = \dfrac{c}{d}$ by $bd$:

$\dfrac{a \cdot bd}{b} = \dfrac{c \cdot bd}{d} \Rightarrow ad = bc$

What it’s for

• Simplify a fraction: if $\dfrac{6}{9} = \dfrac{2}{3}$, you mentally check $6 \times 3 = 9 \times 2$ ✓.
• Find a missing value: $\dfrac{x}{7} = \dfrac{15}{21}$ → $21x = 105$ → $x = 5$.
• Proportionality: it’s exactly the “do these two ratios match?” test.

?Your turn

Are 14/21 and 6/9 equal?

A.YesB.No

Check