Question

The ratio of third proportional to $12$ and $30$ and the mean proportional between $9$ and $25$ is

Answer 1

Step 1: Third proportional and Mean proportional

1. The third proportional of a proportion is the second term of the mean terms. For example, if we have $a:b=c:d$, then the term ‘$c$’ is the third proportional to ‘$a$’ and ‘$b$’.
2. The mean proportional between the two terms of a ratio in a proportional is the square root of the product of these two. For example, in the proportion $a:b::c:d$, we can define the mean proportional for the ratio $a:b$ as the square root of the product of the two terms of the ratio or $\sqrt{ab}$.

Step 2: Find the third proportional

Let the third proportional to $12$ and $30$ be $x$.

$\Rightarrow 12:30::30:x$

$\begin{array}{rcl}& \Rightarrow & 12x=30\times 30\\ & \Rightarrow & x=\frac{(30\times 30)}{12}\\ & \Rightarrow & x=75\end{array}$

Therefore, the third proportional to $12$ and $30$ is $75$.

Step 3: Find the mean proportional

The mean proportional between $9$ and $25$ is $\sqrt{9\times 25}$.

$\begin{array}{rcl}& \Rightarrow & \sqrt{9\times 25}=\sqrt{255}\\ & \Rightarrow & \sqrt{255}=15\end{array}$

Therefore,

Ratio $=75:15$

Ratio $=5:1$

Hence, the ratio of third proportional to $12$ and $30$ and the mean proportional between $9$ and $25$ is $5:1$.