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The ratio of third proportional to 12 and 30 and the mean proportional between 9 and 25 is
A. \[2:1\]
B. \[5:1\]
C. \[7:15\]
D. \[9:14\]

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Hint: In a continued proportion, the third proportional of two numbers \[a\] and \[b\] is defined to be that number \[c\] such that \[a:b = b:c\]. A mean proportional is a number that comes between two other numbers and which satisfies the equation \[\dfrac{a}{x} = \dfrac{x}{b}\].

Let the third proportional be \[x\],
So, the continued third proportion is \[12:30:x\]
i.e., \[12:30 = 30:x\]
\[
   \Rightarrow 12 \times x = 30 \times 30 \\
   \Rightarrow x = \dfrac{{30 \times 30}}{{12}} \\
   \Rightarrow x = \dfrac{{900}}{{12}} \\
  \therefore x = 75 \\
\]
Let the mean proportion be \[y\]
We know that a mean proportional is a number that comes between two other numbers and which satisfies the equation \[\dfrac{a}{x} = \dfrac{x}{b}\].
So, the mean proportion is \[\dfrac{9}{y} = \dfrac{y}{{25}}\]
\[
   \Rightarrow {y^2} = 9 \times 25 \\
   \Rightarrow {y^2} = 225 \\
   \Rightarrow y = \sqrt {225} \\
  \therefore y = 15 \\
\]
Therefore, the ratio of third proportional to 12 and 30 and the mean proportional between 9 and 25 is \[x:y = 75:15 = 5:1\]
Thus, the answer is option B. \[5:1\]

Note: A third proportional is also equal to the square of the second term, divided by the first term. And a mean proportional is equal to the square root of the product of first and second term.
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