Question

Find the ratio compounded of
(1) the ratio $2a:3b$, and the duplicate ratio of $9b^2:ab$.
(2) the subduplicate ratio of $64:9$, and the ratio $27:56$.
(3) the duplicate ratio of $\frac{2a}{b}:\frac{\sqrt{6a^2}}{b^2}$, and the ratio $3ax:2by$.

Answer 1

$1.$

Duplicate ratio of $9b^2:ab={\left(\frac{9b^2}{ab}\right)}^2=\frac{81b^4}{a^2{b}^2}$

The required ratio $=\frac{2a}{3b}\times \frac{81b^4}{a^2{b}^2}$ $=\frac{54b}{a}$

$2.$

The sub-duplicate ratio of $64:9=\sqrt{\frac{64}{9}}=\frac{8}{3}$

The required ratio $=\frac{27}{56}\times \frac{8}{3}$ $=\frac{9}{7}$

$3.$

The duplicate ratio of $\frac{2a}{b}:\frac{\sqrt{6a^2}}{b^2}={(\frac{2a}{b}:\frac{\sqrt{6a^2}}{b^2})}^2$

$=\frac{4a^2}{b^2}:\frac{6a^2}{b^4}$

The required ratio $=\frac{4a^2}{b^2}:\frac{6a^2}{b^4}\times \frac{3ax}{2by}$

$=\frac{4a^2}{b^2}\times \frac{b^4}{6a^2}\times \frac{3ax}{2by}$

$=\frac{bax}{y}$