H S Hall and S R Knight Solutions for Chapter: Ratio, Exercise 1: Examples. I .

Find the compounded ratio of the sub-duplicate ratio of $64:9$ and the ratio $27:56.$

Find the compounded ratio of the duplicate ratio of $\frac{2a}{b}:\frac{\sqrt{6}{a}^2}{b^2}$  and the ratio $3ax:2by.$

If $\frac{l}{\sqrt{a}-\sqrt{b}}+\frac{m}{\sqrt{b}-\sqrt{c}}+\frac{n}{\sqrt{c}-\sqrt{a}}=0$ and $\frac{l}{\sqrt{a}+\sqrt{b}}+\frac{m}{\sqrt{b}+\sqrt{c}}+\frac{n}{\sqrt{c}+\sqrt{a}}=0$, show that $\frac{l}{(a-b)(c-\sqrt{ab)}}=\frac{m}{(b-c)(a-\sqrt{bc})}=\frac{n}{(c-a)(b-\sqrt{ac})}.$

Solve the equations $ax+by+cz=0, bcx+cay+abz=0, xyz+abc\left(a^{3}x+b^{3}y+c^{3}z\right)=0.$

Solve the equations $ax+by+cz=a^{2}x+b^{2}y+c^{2}z=0$ and $x+y+z+(b-c)(c-a)(a-b)=0.$

If $a(y+z)=x, b(z+x)=y, c(x+y)=z$, prove that $\frac{x^2}{a(1-bc)}=\frac{y^2}{b(1-ca)}=\frac{z^2}{c(1-ab)}$.

If $ax+hy+gz=0, hx+by+fz=0, gx+fy+cz=0$, prove that $\frac{x^2}{bc-f^2}=\frac{y^2}{ca-g^2}=\frac{z^2}{ab-h^2}.$

If $ax+hy+gz=0, hx+by+fz=0, gx+fy+cz=0$, prove that $\left(bc-f^2\right)\left(ca-g^2\right)\left(ab-h^2\right)=(fg-ch)(gh-af)(hf-bg).$