Solve each problem. Suppose two acid solutions are mixed. One is ...

Question

Solve each problem. Suppose two acid solutions are mixed. One is 26% acid and the other is 34% acid. Which one of the following concentrations cannot possibly be the concentration of the mixture? A. 24% B. 30% C. 31% D. 33%

Accepted Answer

Welcome back. I am so glad you're here. We are given this model application problem. There are two aromatic solutions with respective concentrations of 37% and 59%, which among the following cannot be the final concentration of the mixture. So we're envisioning we've got two aromatic solutions, one of them Is 37% concentrated, the other is 59% concentrated. We are going to combine them in some way. We don't know how much of the quantities of each. And we want to know which, what's the range of the final concentration of the mixture. So I'm going to explain this in words, and then I'm going to explain it algebraic Lee. So we know that when we pour any combination of these two aromatic solutions, the range is going to be from the lowest, which is 37% to the highest, which is 59%. It has to be within that range. It can't get any lower, it can't get any higher if that makes sense to you, then great. I'm sure you can discern the answer choice if that's still a little bit confusing. Let me explain it. Algebraic lee. So we recall from previous lessons that for mixture problems, we are going to take the strength in one column and we're going to take the amount of measurement, I'm just going to say leaders, but it could be any form of measurement in another column. And then we're going to have the leaders or whatever the measurement is of the mixture in the final column. And so the first row is going to be our first Strength percentage. So here that's 37%. The second row Is going to be the second strengths percentage, which is 59%. And the third row is going to be our mixtures percentage. And that's what we're trying to find. We don't know what it is. So I'm going to call that Z Now the leaders, that's the quantity, how much of each thing. And we don't actually know what our quantities are for this one. So in this one, we can call the quantity of the 37%, we can call the quantity of the 59%. Why? And the quantity of Z of our final mixture is going to be X plus Y. And to find how many leaders of the mixture there are, we multiply straight across these first two columns. So it's going to be I'm going to change 37% to a decimal. So it's going to be 370.37 X for the first one, for the second one. It's going to be 10.59 Y and for the mixture, it's going to be easy times parentheses X plus Y. Now this still might be confusing but stick with me. So in order to find out to solve for R Z, what we would do is we would say okay, the first two added together of this third column, this 20.37 X plus the 0.59 Y is going to add up to the third column, third row here, it'll add up to Z times X plus Y. And now we can say that since we don't know what quantities, how many leaders of each strength of aromatic solution is going in. We could say, okay, what if we throw in nothing of the first one? What if we throw in no parts of that, that's zero. And we could assign any quantity, but let's just say one for the 59% solution and that would equal still Z that's what we're looking for. And we know that X is zero plus Y is one and now we can solve for Z. So if we weren't putting in any of the 37% and we only put in the 59%. How, what are we going to get for our final mixture? Well, 37 is 0.59 times one is 10. equals Z times zero plus one is Z times one, which is just Z. So if we put in the maximum amount of the 59% and the minimum amount of the 37%, which is not none of it. We're still only going to get the highest amount being 59%. We could do the same thing. Plugging in zero for the 59%, see what the lowest we would get is. So for 37%, let's say we're putting in One quantity of that one leader here for this specific example. And for the 59%, we're putting in none of it. And then that's going to equal RZ that we're looking for times this time X is one plus this time Y is zero. And on the left 0.37 times one is our 10. and 0.59 times zero is none of that 59%. And then in the parentheses, one plus zero is one and one time Z is Z. So here, The other end of the spectrum gives us the lowest amount we can have is 37% for liquid. So it's going to be between 37% and 59%. So now we look at our answer choices. It needs to be between And 59%. And we can rule out answer choice C that falls outside of the range. And what we're looking for is the one that cannot be. So answer choice C tells us which cannot be the final concentration of the mixture. Well done. We'll catch you on the next one.

Solve each problem. Suppose two acid solutions are mixed. One is 26% acid and the other is 34% acid. Which one of the following concentrations cannot possibly be the concentration of the mixture? A. 24% B. 30% C. 31% D. 33%

Key Concepts

Weighted Average

Concentration Range

Linear Combination

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