<p><span style="color:#27ae60;"><span style="font-size:20px;">Solving the Problem</span></span></p>
<p>This seems like a very wordy question, so it's best if we tackle it <strong><span style="color:#8e44ad;">Piece by Piece</span></strong>. We can start by reading and making deductions:</p>
<blockquote>
<p>A tailor used 30 buttons that had an average (arithmetic mean) weight of x grams per button</p>
</blockquote>
<p>We can start by determining the sum of these 30 buttons:</p>
<ul>
<li>$sum = mean * \#\,values$</li>
<li>$sum\,of\,30\,buttons = x * 30$</li>
<li>The sum of these buttons will be $30x$</li>
</ul>
<blockquote>
<p>and 20 other buttons that had an average weight of 80 grams per button.</p>
</blockquote>
<p>Now, let's find the sum of the remaining 20 buttons:</p>
<ul>
<li>$sum = mean * \#\,values$</li>
<li>$sum\,of\,20\,buttons = 80 * 20$</li>
<li>The sum of these buttons will be $1600$</li>
</ul>
<blockquote>
<p>Which of the following is the average weight per button, in grams, of the 50 buttons that the tailor used?</p>
</blockquote>
<p>Finally, we can work out the sum and then the mean of all 50 buttons:</p>
<ul>
<li>$mean\,of\,50\,buttons = \frac{sum\,of\,50\,buttons}{50}$</li>
<li>$sum\,of\,50\,buttons = 30x + 1600$</li>
<li>$mean\,of\,50\,buttons = \frac{30x + 1600}{50}$</li>
<li>$mean\,of\,50\,buttons = \frac{3}{5}x + 32$</li>
</ul>
<p>So, the answer will be <span style="color:#27ae60;">D</span>.</p>