At a math convention, a poll is taken about cable news. It turns out that $7$ out of every $12$ people are bored by cable news. If there are $280$ people at the convention who are bored by cable news, then how many people at the convention are not bored by cable news?
We want to find the total number of people at the convention, not just the ones who are bored by cable news. Let $x$ be the total number of people at the convention. Then we know that $\frac{7}{12}$ of them are bored by cable news, so we can set up the equation:
$$\frac{7}{12}x = 280$$
To solve for $x$, we can multiply both sides by $\frac{12}{7}$:
$$x = 240\cdot\frac{12}{7} = \frac{2880}{7} \approx 411$$
So there are approximately $411$ people at the convention in total.
To find the number of people who are not bored by cable news, we can subtract the number of people who are bored by cable news from the total:
$$411 - 280 = 131$$
Therefore, there are $131$ people at the convention who are not bored by cable news.
To find the number of people at the convention who are not bored by cable news, we need to determine the total number of people at the convention.
Let's assume the total number of people at the convention is $x$.
According to the given information, $\frac{7}{12}$ of the people are bored by cable news.
Therefore, the number of people bored by cable news is $\frac{7}{12} \cdot x$.
We are given that this number is $280$, so we can write the equation:
$\frac{7}{12} \cdot x = 280$
To solve for $x$, we can multiply both sides of the equation by $\frac{12}{7}$:
$x = 280 \cdot \frac{12}{7}$
$x = 480$
Therefore, there are $\boxed{480}$ people at the convention who are not bored by cable news.