Cross multiplication in equations

When two fractions are set equal, $\frac{a}{b} = \frac{c}{d}$, cross-multiplying gives $a \times d = b \times c$. You multiply the numerator of each side by the denominator of the other side. This removes the fractions in one step.

Take $\frac{x}{6} = \frac{2}{3}$. Cross-multiplying gives $3x = 6 \times 2 = 12$, so $x = 4$. There is no need to find a common denominator first.

A ratio written with a colon is just a fraction, so the same trick works. For instance $x : 150 = 3 : 5$ is the same as $\frac{x}{150} = \frac{3}{5}$, and cross-multiplying gives $5x = 150 \times 3$.

Cross-multiplication also clears a variable in the denominator. From $\frac{y^2}{y + 2} = \frac{49}{9}$ you get $9y^2 = 49(y + 2)$, which you can then expand into a standard quadratic.

Solve $\frac{x}{4} = \frac{9}{6}$.
Cross-multiply: $6x = 4 \times 9 = 36$.
So $x = 6$.

Solve the ratio equation $x : 120 = 2 : 3$.
Write it as $\frac{x}{120} = \frac{2}{3}$ and cross-multiply: $3x = 120 \times 2 = 240$.
So $x = 80$.

Turn $\frac{y^2}{y - 1} = \frac{16}{5}$ into a standard quadratic.
Cross-multiply: $5y^2 = 16(y - 1) = 16y - 16$.
So $5y^2 - 16y + 16 = 0$.