Find the ratio between the fourth proportional of 1, 3 and 6 to the mean proportional of 12 and 3.

Step 1: Find the fourth proportional

If numbers are $a, b, c$, then the fourth proportional $d$ is given by

$d = \frac{b \cdot c}{a}$

Here, $a = 1$, $b = 3$, $c = 6$

$d = \frac{3 \cdot 6}{1} = 18$

So, the fourth proportional is $18$.

Step 2: Find the mean proportional

If numbers are $x$ and $y$, the mean proportional $m$ is

$m = \sqrt{x \cdot y}$

Here, $x = 12$, $y = 3$

$m = \sqrt{12 \cdot 3} = \sqrt{36} = 6$

Step 3: Find the ratio

The ratio of fourth proportional to mean proportional is

$\frac{18}{6} = 3 : 1$

✅ Answer:

3 : 1