Solving the Problem

First of all, what kind of problem is this? It's a Multiple-Select problem because test takers can select at least one option, and in this case, up to five options. These questions are terrible for guessing. The probability you guess the correct answer for this problem is $1$ out of $31$. So...yeah, guessing is not a good idea.

How do we solve it then? In many multiple-select problems, the goal is to find the boundaries: the minimum possible value and the maximum possible value. You then also assume that all values between those two are valid as well.

Calculating the minimum gross income value:

If we set up the following equations, we can calculate Kate's minimum gross income:

$\frac{1}{3} \times gross \: income = 13$,$470$

$gross \: income = 3 \times 13$,$470 = 40$,$410$

Calculating the maximum gross income value:

If we set up the following equations, we can calculate Kate's maximum gross income:

$\frac{1}{4} \times gross \: income = 13$,$470$

$gross \: income = 4 \times 13$,$470 = 53$,$880$

Now we can say that Kate's gross income conforms to the following inequality:

$40$,$410 \leq gross \: income \leq 53$,$880$

Thus, the correct answers are B, C, D, and E.