x and y are two numbers such that their mean proportion is 9 and third proportion is 243. What is the value of x and y?

Three numbers a,b,c are in proportion iff $$b^2 = ac$$ where $$b$$ is the mean proportion and $$c$$ is the third proportion

Mean proportion of two numbers $$x$$ and $$y$$ = 9

=> $$xy = (9)^2 = 81$$ ---------------(i)

Third proportion = 243

=> $$y^2 = x \times 243$$ -------------(ii)

Substituting value of $$x$$ from equation(i) in equation(ii), we get :

=> $$y^2 = \frac{81}{y} \times 243$$

=> $$y^3 = (3)^4 \times (3)^5 = (3)^{9}$$

=> $$y = (3)^{\frac{9}{3}} = 3^3 = 27$$

Substituting it in equation(i), => $$x = \frac{81}{27} = 3$$

=> Ans - (B)