The Mixture Trick

Note: 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"). The Mixture Trick 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. Example 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? The Trick Step 1: Put the lower percentage on the left, the combined percentage in the middle, and the higher percentage on the right. $$30..........................38..........................42$$ Step 2: Calculate the "gap" or "distance" between each number and the middle number. $$30..........(8)................38.............(4).............42$$ Step 3: 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. So the final ratio of Fabulous Soil to BackYard Dirt is $1$:$2$, NOT $2$:$1$.