This question requires us to find the ratio between two specific mathematical values: the fourth proportional of a given set of numbers and the mean proportional of another pair of numbers.

Finding the Fourth Proportional

We are given the numbers 2, 5, and 4. Let the fourth proportional be '$x$'. The definition of a fourth proportional means that the ratio between the first two numbers is equal to the ratio between the third and fourth numbers.

We can set up the proportion like this:

$$ 2 : 5 :: 4 : x $$

This can be written as a fraction equation:

$$ \frac{2}{5} = \frac{4}{x} $$

To find '$x$', we can cross-multiply:

$$ 2 \times x = 5 \times 4 $$

$$ 2x = 20 $$

Now, we solve for '$x$' by dividing both sides by 2:

$$ x = \frac{20}{2} $$

$$ x = 10 $$

So, the fourth proportional is 10.

Calculating the Mean Proportional

Next, we need to find the mean proportional between 2.5 and 0.016. The mean proportional (or geometric mean) between two numbers '$a$' and '$b$' is given by the formula $\sqrt{a \times b}$.

In this case, $a = 2.5$ and $b = 0.016$.

First, let's multiply these two numbers:

$$ 2.5 \times 0.016 $$

It might be easier to work with fractions:

$$ 2.5 = \frac{25}{10} = \frac{5}{2} $$

$$ 0.016 = \frac{16}{1000} $$

Now multiply them:

$$ \frac{5}{2} \times \frac{16}{1000} = \frac{5 \times 16}{2 \times 1000} = \frac{80}{2000} $$

Simplify the fraction:

$$ \frac{80}{2000} = \frac{8}{200} = \frac{1}{25} $$

Now, find the square root of this result to get the mean proportional:

$$ \sqrt{\frac{1}{25}} = \frac{1}{5} $$

As a decimal, the mean proportional is:

$$ \frac{1}{5} = 0.2 $$

So, the mean proportional is 0.2.

Determining the Ratio

The question asks for the ratio of the fourth proportional (which is 10) to the mean proportional (which is 0.2).

The ratio is:

$$ 10 : 0.2 $$

To express this ratio using whole numbers, we can multiply both parts of the ratio by 10 to eliminate the decimal:

$$ (10 \times 10) : (0.2 \times 10) $$

$$ 100 : 2 $$

Finally, simplify this ratio by dividing both parts by their greatest common divisor, which is 2:

$$ \frac{100}{2} : \frac{2}{2} $$

$$ 50 : 1 $$

Therefore, the ratio of the fourth proportional to the mean proportional is 50 to 1.