Percent is just figure out what proportion of $100$ your quantity is. What's $50\%$ of 30? It's whatever value over $30$ is equal in proportion to $50$ over $100$. So in the first case, you want to solve for
$\frac{27}{408} = \frac{x_1}{100}$ and in the other case $\frac{37}{582} = \frac{x_2}{100}$. Contrary to the other answer, percent is not always the smaller number divided by the bigger number. For example, $150$ is $150\% $ of $100$, not $66.7\%$.
To figure out how to get the percents equal to the first case, you solve
$\frac{37+n}{582 + n} = \frac{x_1}{100} = \frac{27}{408}$
because if you add only the word fertility $n$ times to your new essay, you'll have the 37 old ones plus the $n$ new fertility. The total number of words is similarly $582 + n$.
If it turns out that $n$ is negative when you solve this, that means the percent of fertility in the new essay is actually more than the percent in the old one! This means you'd need to remove fertility a few times to make the percents equal.
If this is still confusing to you, please do not hesitate to ask.
