$6$ is the mean proportion between two numbers $x$ and $y$, and $48$ is the third proportional to $x$ and $y$. Find the numbers.
$6$ is the mean proportion between $x$ and $y$.
$\implies$ $x$, $6$ and $y$ are in continued proportion.
$\implies$ $x : 6 = 6 : y$
$\implies$ $6^2 = x \times y$ $\implies$ $36 = xy$ $\;\;\; \cdots \; (1)$
$48$ is the third proportional to $x$ and $y$.
$\implies$ $x$, $y$ and $48$ are in continued proportion.
$\implies$ $x : y = y : 48$
$\implies$ $y^2 = 48 x$ $\;\;\; \cdots \; (2)$
i.e. $\;$ $y^2 = 48 \times \dfrac{36}{y}$ $\;\;\;$ [substituting the value of $x$ from equation $(1)$]
i.e. $\;$ $y^3 = 1728 = 12^3$ $\implies$ $y = 12$
Substituting the value of $y$ in equation $(1)$ gives
$x = \dfrac{36}{y} = \dfrac{36}{12} = 3$