Cross Products

In a proportion $\frac{a}{b} = \frac{c}{d}$, multiply across the equals sign: the top left by the bottom right, and the bottom left by the top right. These cross products are always equal: $a\cdot d = b\cdot c$. In $\frac{3}{4} = \frac{9}{12}$, the cross products are 3 · 12 and 4 · 9, and both are 36.

Why does it work? Give both fractions the same bottom, 4 · 12 = 48. Then $\frac{3}{4}$ is 36 over 48 and $\frac{9}{12}$ is also 36 over 48. The new tops are exactly the cross products, 3 · 12 and 9 · 4. Equal fractions with the same bottom must have the same top, so the cross products match.

That makes cross products a quick test of whether two ratios are equal, with no simplifying. Are $\frac{4}{6}$ and $\frac{10}{15}$ equal? 4 · 15 is 60 and 6 · 10 is 60, so yes. Are $\frac{3}{4}$ and $\frac{5}{7}$? 3 · 7 is 21 but 4 · 5 is 20, so no.