Question

# 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. $9:14$

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.