To solve this problem, we can set up an equation using the given information. Let`s assume the number we are trying to find is "x".

According to the problem, 35% of the number is 416 more than 27% of the same number. Mathematically, this can be written as:

0.35x = 0.27x + 416

To solve for x, we can subtract 0.27x from both sides of the equation:

0.35x - 0.27x = 416

Simplifying this equation gives us:

0.08x = 416

Now, to isolate x, we divide both sides of the equation by 0.08:

x = 416 / 0.08

Performing the calculation, we find that x is equal to 5200.

Therefore, the correct answer is option 1: 5200.

This means that the number we were trying to find is 5200.