Let the required numbers be a and b.
Given, 14 is the mean proportional between a and b.
![\[\begin{array}{*{35}{l}} a:\text{ }14\text{ }=\text{ }14:\text{ }b \\ ab\text{ }=\text{ }196 \\ a\text{ }=\text{ }196/b\text{ }\ldots .\text{ }\left( 1 \right) \\ \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-34daa68f6269a34a4ded4fb13496889b_l3.png "Rendered by QuickLaTeX.com")
Also, given, third proportional to a and b is 112.
![\[\begin{array}{*{35}{l}} a:\text{ }b\text{ }=\text{ }b:\text{ }112 \\ {{b}^{2}}~=\text{ }112a\text{ }\ldots .\text{ }\left( 2 \right) \\ \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-008316767fa26bf69a211e4669f21199_l3.png "Rendered by QuickLaTeX.com")
Using (1), we have:
![\[\begin{array}{*{35}{l}} {{b}^{2}}~=\text{ }112\text{ }\times \text{ }\left( 196/b \right) \\ {{b}^{3}}~=\text{ }{{14}^{3}}~x\text{ }{{2}^{3}} \\ b\text{ }=\text{ }28 \\ \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-c490753fa830c7a196bfa3960ddddf7a_l3.png "Rendered by QuickLaTeX.com")
From (1),
a = 196/ 28 = 7
Therefore, the two numbers are 7 and 28.
Find two numbers such that the mean proportional between them is 14 and third proportional to them is 112.
Find two numbers such that the mean proportional between them is 14 and third proportional to them is 112.
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