Question : Two numbers $x$ and $y$ are such that their mean proportion is 9 and the third proportion is 243. What are the values of $x$ and $y$?
Option 1: 3 and 9
Option 2: 3 and 27
Option 3: 6 and 27
Option 4: 6 and 81
Correct Answer: 3 and 27
Solution :
Two given numbers are $x$ and $y$.
The mean proportion between $x$ and $y$ is 9.
⇒ $x:9=9:y$
⇒ $\frac{x}{9}=\frac{9}{y}$
$⇒x=\frac{81}{y}$ ------------------------------(1)
The third proportion to $x$ and $y$ is 243.
⇒ $x:y=y:243$
⇒ $\frac{x}{y}=\frac{y}{243}$
By putting the value of $x$, we get,
$⇒y^{2}=243×\frac{81}{y}$
$⇒y^{3}=243×81$
$⇒y^{3}=(27)^{3}$
$\therefore y=27$
Putting this value in equation (1), we get,
$\therefore x=\frac{81}{27}=3$
Hence, the correct answer is 3 and 27.
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