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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) \[9:14\]

Answer
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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  \[\dfrac{a}{x} = \dfrac{x}{b}\].

Complete step by step solution:
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 terms.