Question

Marie mixes 16 liters of 18% acid solution with a 27% acid solution to make a 21% acid solution.

How many liters of 27% solution did she use?

Answer 1

So I'm going to show you the idea behind it, so that it makes sense to you, and hopefully you can solve it on your own from then on, on tests, or other homeworks, or whatever. So let's get started!

So first off, think about the units. Similar to how if you drive a car at 60 miles/hour for 1 hour, you have: \[60\frac{ miles }{ hour }*1 hour=60 miles\] since the units cancel out, an hour/hour is just =1, so we are just left with miles. Similarly if you're given:

16 Liters of 18% acid solution. What does this mean? It means you have .18 acid/liter. So this is going to be kind of a big jump, but can you take a guess at a possible way to relate all of these together with one variable, call it V, that represents how much is used of the .27 acid solution.

Answer 2

\[(.18\frac{ acid }{ Liters }*16Liters)+(.27\frac{ acid }{ Liters }*V)=.21\frac{ acid }{ Liters }*(16Liters+V)\]

Maybe try to think what this represents. I came up with this because I can keep track of the amount of acid this way easier, since I'm just adding the plain acid amounts.

See, the left side has two terms added together. The amount of acid per liter and the amount of liters. Just like the distance=rate*time I did earlier. The other term with the V in it on the left side of the equals sign shows how much acid per liter, but we don't know what the volume should be, so we leave it as a variable to solve for.

So now on the right side of the equals we have .21 acid/Liters and this time since it's a combination of both liquids, we add the 16 Liters along with V since they're being mixed. Seems kind of magical, and it is, math is awesome!, and what you have here is something that's equal because the total amount of acid on the left is the same as on the right. Now you can solve for V, problem solved!