Question

Find the fourth proportional of the following:

(1) 14, 21, 4
(2) a², ab, b²
(3) 5, $\sqrt{75},\sqrt{48}$

Answer 1

(1)
$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{fourth}\mathrm{proportional}\mathrm{be}x. \\ \therefore \frac{14}{21}=\frac{4}{x} \\ \Rightarrow 14x=84 \\ \Rightarrow x=6 \end{gathered}$$

(2)
$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{fourth}\mathrm{proportional}\mathrm{be}x. \\ \therefore \frac{a^2}{ab}=\frac{b^2}{x} \\ \Rightarrow a^{2}x=a{b}^3 \\ \Rightarrow x=\frac{b^3}{a} \end{gathered}$$

(3)
$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{fourth}\mathrm{proportional}\mathrm{be}x. \\ \therefore \frac{5}{\sqrt{75}}=\frac{\sqrt{48}}{x} \\ \Rightarrow 5x=4\sqrt{3}\times 5\sqrt{3} \\ \Rightarrow 5x=60 \\ \Rightarrow x=12 \end{gathered}$$