A bag contains yellow, green, blue, and white marbles. The ratio of yellow, green, blue, white marbles is $4:3:2:1$. If the bag contains $187$ marbles, then how many blue marbles are there?
Let the total number of marbles be $x$. Then, we have the following system of equations: $$\frac{4}{10}x + \frac{3}{10}x + \frac{2}{10}x + \frac{1}{10}x = x = 187.$$ Simplifying and solving for $x$, we get $x=1870$. So, there are $\frac{2}{10}\cdot 1870 = \boxed{374}$ blue marbles.
To solve this problem, we'll start by finding the total number of parts in the ratio.
The ratio of yellow, green, blue, and white marbles is $4:3:2:1$.
Adding up all the parts, we have $4 + 3 + 2 + 1 = 10$ parts.
Next, we'll find the value of each part by dividing the total number of marbles by the number of parts.
Since the bag contains a total of $187$ marbles, each part is equal to $187/10 = 18.7$ (approx).
Since we cannot have a fraction of a marble, we'll round this down to the nearest whole number, which is $18$.
Therefore, there are $18$ blue marbles in the bag.