<p><strong>Note</strong>: in the second example, we say that the amount of solution x is 11/3 times larger than that of solution y. It's actually just 8/3 times larger than solution y (i.e, x is 11/3 times than that of solution y - notice the importance of the word "larger"). </p><p><span style="color:#27ae60;">The Mixture Trick</span> is really frickin' cool. It's a shortcut that allows you to solve these problems in a flash. It's best to demonstrate how it works with an example.</p>
<p><strong><span style="color:#8e44ad;">Example</span></strong></p>
<p>Bill mixes Fabulous Soil ($30\%$ nitrogen by volume) with BackYard Dirt ($42\%$ by volume) and finds that the combined soil contains $38\%$ nitrogen by volume. In what ratio were the two soils mixed?</p>
<p><strong><span style="color:#8e44ad;">The Trick</span></strong></p>
<p style="margin-left: 40px;"><strong>Step 1</strong>: Put the lower percentage on the left, the combined percentage in the middle, and the higher percentage on the right.</p>
<p>$$30..........................38..........................42$$</p>
<p style="margin-left: 40px;"><strong>Step 2</strong>: Calculate the "gap" or "distance" between each number and the middle number.</p>
<p>$$30..........(8)................38.............(4).............42$$</p>
<p style="margin-left: 40px;"><strong>Step 3</strong>: The two "distance" numbers we found in step 2 represent the initial mixing ratio. So that's $8$:$4$, or $2$:$1$. But which one is $2$ and which one is $1$? Well, notice how the $38$ number is CLOSER to the $42$ number. That means that there must be MORE of the $42\%$ mixture, so we actually have to "flip" the ratio for it to make sense.</p>
<p>So the final ratio of Fabulous Soil to BackYard Dirt is <span style="color:#27ae60;">$1$:$2$</span>, <span style="color:#e74c3c;">NOT $2$:$1$</span>.</p>