Find the ratio between the fourth proportional of 8, 12 and 4, and the third proportional of 6 and 9.

The correct option is 4:9.

To find the required ratio, we need to calculate two values:
1. The fourth proportional of 8, 12, and 4.
2. The third proportional of 6 and 9.

Step 1: Find the fourth proportional of 8, 12, and 4.

Let the fourth proportional be $x$.

By definition, if four numbers $a$, $b$, $c$, and $d$ are in proportion, then $a:b=c:d$. Therefore, we can set up the proportion:

$8:12=4:x$

This can be written as a fraction:

$\frac{8}{12}=\frac{4}{x}$

By simplifying the fraction, we get:

$\frac{2}{3}=\frac{4}{x}$

Cross-multiplying gives:

$2x=12$

Solving for $x$ gives:

$x=6$

Thus, the fourth proportional is 6.

Step 2: Find the third proportional of 6 and 9.

Let the third proportional be $y$.

By definition, if three numbers $a$, $b$, and $c$ are in continued proportion, then $a:b=b:c$. Therefore, we have:

$6:9=9:y$

This can be written as:

$\frac{6}{9}=\frac{9}{y}$

Simplifying the fraction gives:

$\frac{2}{3}=\frac{9}{y}$

Cross-multiplying to solve for $y$ gives:

$2y=27$

Solving for $y$ gives:

$y=\frac{27}{2}$

Thus, the third proportional is $\frac{27}{2}$.

Step 3: Find the ratio between the fourth proportional and the third proportional.

We need to find the ratio $x:y$:

$x:y=6:\frac{27}{2}$

To simplify this ratio, we multiply both terms by 2:

$12:27$

Dividing both sides of the ratio by their greatest common divisor, which is 3, we get:

$4:9$

Therefore, the ratio is 4:9, confirming that the correct answer is indeed 4:9.