Question
Find the two numbers such that their mean proprtional is $24$ and the third proportinal is $1,536.$
✓
Answer
Let x and y be two numbers
Mean proportional $= 24$
$\Rightarrow \sqrt{x y}=24 $
$ \Rightarrow x y=24 \times 24=576$
$\Rightarrow x=\frac{576}{y} ...(1)$
Also $1536$ is the third proportional then
$x : y = y : 1,536$
$\Rightarrow \frac{x}{y}=\frac{y}{1,536} $
$ \text { From(1), } y ^2=1,536 \times \frac{576}{y} $
$ \Rightarrow y ^3=1,536 \times 576 $
$ \Rightarrow y ^3=24 \times 24 \times 24 \times 24$
$\Rightarrow y=24 \times 24 $
$ \Rightarrow y=96$
Again form $(1),$ we get
$x=\frac{576}{96} $
$=6.$
Hence, the required numbers are $6$ and $$96.$
Mean proportional $= 24$
$\Rightarrow \sqrt{x y}=24 $
$ \Rightarrow x y=24 \times 24=576$
$\Rightarrow x=\frac{576}{y} ...(1)$
Also $1536$ is the third proportional then
$x : y = y : 1,536$
$\Rightarrow \frac{x}{y}=\frac{y}{1,536} $
$ \text { From(1), } y ^2=1,536 \times \frac{576}{y} $
$ \Rightarrow y ^3=1,536 \times 576 $
$ \Rightarrow y ^3=24 \times 24 \times 24 \times 24$
$\Rightarrow y=24 \times 24 $
$ \Rightarrow y=96$
Again form $(1),$ we get
$x=\frac{576}{96} $
$=6.$
Hence, the required numbers are $6$ and $$96.$
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