If $80\%$ of the people in the science club of $15$ people are in the Math Club, $\frac{4}{5}\times15=12$ people are in the both the Math Club and the Science Club.
These $12$ people make up $30\%$ of the Math Club.
Setting up a proportion:
$12\cdot 1.00 =0.30\cdot x$
$\frac{12}{0.3} = 40=x$
There are $40=\boxed{E}$ people in the Math Club.

To determine the number of students in the Math Club, we start by analyzing the given information:

1. $30\%$ of the students in the Math Club are also in the Science Club.
2. $80\%$ of the students in the Science Club are also in the Math Club.
3. There are $15$ students in the Science Club.

Let's denote the number of students in the Math Club as $M$.

According to the problem, $30\%$ of the students in the Math Club are in the Science Club. Therefore, the number of students in both the Math Club and the Science Club is $0.3M$.

We are also given that $80\%$ of the students in the Science Club are in the Math Club. Since there are $15$ students in the Science Club, the number of students in both the Science Club and the Math Club is $0.8 \times 15 = 12$.

Since the number of students in both clubs is the same, we can set up the following equation:
$$0.3M = 12$$

To find $M$, we solve for $M$ by dividing both sides of the equation by $0.3$:
$$M = \frac{12}{0.3} = \frac{120}{3} = 40$$

Therefore, the number of students in the Math Club is $\boxed{\text{E}}$.