🟰 Proportion

• Cross-multiplication: in $a:b=c:d$ the outer terms $a$ and $d$ are the extremes and the inner terms $b$ and $c$ are the means. The rule is product of extremes = product of means: $a\times d=b\times c$.

• Fourth proportional: given three numbers $a,b,c$, the fourth proportional $d$ satisfies $a:b=c:d$, so $d=\dfrac{b\times c}{a}$.

• Third proportional: given two numbers $a,b$, the third proportional repeats the middle term — $a:b=b:x$ — giving $x=\dfrac{b^2}{a}$.

• Mean proportional: the single value $x$ sitting between $a$ and $b$ with $a:x=x:b$. Then $x^2=a\times b$, so $x=\sqrt{a\times b}$ — the geometric mean, not the ordinary average.

• Word problems: decide whether the link is direct (both move the same way — use $\dfrac{x_1}{y_1}=\dfrac{x_2}{y_2}$) or inverse (they move opposite ways — use $x_1 y_1=x_2 y_2$), then cross-multiply and solve.