Of four numbers, the second number is $20\%$ more than the first, the third is $10\%$ less than the second and the fourth number is $50\%$ more than the third. If the fourth number is $405$, find all the numbers.

Let the first number be $x$.
The second number is $20\%$ more than the first:

$$\text{Second Number}=1.2x$$

The third number is $10\%$ less than the second:

$$\text{Third Number}=1.2x\times 0.9=1.08x$$

The fourth number is $50\%$ more than the third:

$$\text{Fourth Number}=1.08x\times 1.5=1.62x$$

Given that the fourth number is $405$:

$$1.62x=405$$

$$x=\frac{405}{1.62}=250$$

Now we find all the numbers:

* First number = $250$
* Second number = $1.2\times 250=300$
* Third number = $1.08\times 250=270$
* Fourth number = $405$

Thus, the numbers are $250,300,270,405$.