x and y are two numbers such that their mean proportion is 6 and third proportion is 48. 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 $$y$$ = 6

=> $$xy = (6)^2 = 36$$ ---------------(i)

Third proportion = 48

=> $$y^2 = x \times 48$$ -------------(ii)

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

=> $$y^2 = \frac{36}{y} \times 48$$

=> $$y^3 = (2)^3 \times (6)^3 = (12)^{3}$$

=> $$y = (12)^{\frac{3}{3}} = 12$$

Substituting it in equation(i), => $$x = \frac{36}{12} = 3$$

=> Ans - (D)