🔗 Compound & Chained Ratios

• Compound ratio: $a:b$ compounded with $c:d$ is $(a\times c):(b\times d)$ — multiply the front terms together and the back terms together. Cancel common factors across the ratios first to keep the numbers small.

• Special names: the duplicate ratio of $a:b$ is $a^2:b^2$ (square each term), the sub-duplicate is $\sqrt{a}:\sqrt{b}$, and the triplicate is $a^3:b^3$. “Duplicate” means square, not double.

• Chain method: to combine $A:B$ and $B:C$ into $A:B:C$, make the shared letter $B$ the same number in both.

• Take $A:B=2:3$ and $B:C=4:5$. The value of $B$ is 3 in the first and 4 in the second; their LCM is 12. Scale each so $B=12$: multiply $A:B$ by 4 to get $8:12$, and $B:C$ by 3 to get $12:15$. Stitch: $A:B:C=8:12:15$.

• Shortcut: if you only need the first-to-last term (like $A:D$ in a longer chain), just multiply the ratios like fractions and skip building the full combined ratio.