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Solving Mixture Problems using Weighted Average

Solving Mixture Problems using Weighted Average

Author: Anthony Varela
Description:

Use weighted average to calculate the concentration of a mixed solution.

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Hi, my name is Anthony Varela, and today I'm going to be talking about mixture problems and weighted average. So first we're going to talk about average, how to calculate average. Then we're going to talk about calculating weighted average, and then we're going to use weighted average to solve a mixture problem. So let's review calculating average. And we've all been in this situation before where we've taken three tests in a class, and we want to find our average test score. How would we do that?

Well, to find our average score, we would add up all of the points that we've earned between these three tests, and then we would divide that by the number of tests that we took. So adding up 72, 91, and 83 gives us 246, and then we divide that by 3 to get an average score of 82. So what that means is that scoring a 72, a 91, and an 83 is the same as scoring an 82, an 82, and an 82. So to calculate our average, we add up all of our data, or take the sum of the data, and divide that by the number of data points. In this case, it was three.

So now, what's weighted average? Well, I'm going to use another school or a course example, and I think this is pretty common in lots of course is where you might have your test score which accounts for 1/3 of your grade and then you might have your paper score that accounts for 2/3 of your grade. So here we see that tests and papers are not equally important in terms of your grade. It's actually more important that you score well on your papers than on your tests because that accounts for a bigger percentage of your overall grade.

So let's go ahead and assign some numbers to our test score and our paper score. So your test score is 76 and your paper score is 94. How can we calculate your weighted average? It's not as simple as just adding 76 and 94 and dividing by 2, because papers accounts for more than your tests in your overall grade. So to calculate then the weighted average, we're going to be multiplying these scores by its weight. And so because tests account for 1/3 of your grade, we're going to multiply 76 by 1.

We're just going to count for it once. And because papers account for 2/3 of your grade, we're going to be multiplying 94 by 2. So we're counting that paper grade twice in our calculation for weighted average. And then we need to divide this not by 2, but we're going to divide this by 1 plus 2. And you can think of this as being 1/3 and 2/3 equaling 1 complete course.

So when we go ahead and calculate then this weighted average, our numerator simplifies to 264. That's just adding 76 plus 2 times 94, and then our denominator is 3. So our weighted average then is 88, and if you'd like, you can compare this to our simple average. If you just found the average between 76 and 94, you would get 85. So it's different. This weighted average is different. Our paper score, it was weighted heavier than the test score that influenced our weighted average.

So now we're going to apply this concept of weighted average to solve a classic mixture problem. So I have a 20% sugar concentration by volume, and I'm going to mix that with a 50% sugar concentration. And when I combine these two, what I want as a result, I want 2 quarts of a solution, and I want that to be 30% sugar by volume. So to set this up using weighted average, I'm going to first write x times 0.2. So 0.2 comes from our 20% concentration, so x then represents the quarts of this 20% solution.

I'm going to add to that y time 0.5. So 0.5 comes from our 50% concentration, so y represents the number of quarts of the 50% solution. Now, I'm going to divide this by our total quantity, that would be x plus y, and I know that x plus why we want to equal 2, and we want these 2 quarts to have a concentration of 30% sugar. So this is all going to equal 0.3. That would be 30% written as a decimal.

What if I were to then just do a 50-50 split? So I'm going to put 1 quart of my 20% and 1 quart of my 50%. What would that give us? Well, I'm just going to replace then 1 for x and 1 for y and let's go ahead and solve this fraction or evaluate this fraction. Well, I'm just going to add 0.2 plus 0.5 since I'm multiplying both of them by 1 and then I'm going to divide that by 2, or 1 plus 1. And you can see here then that 0.7 divided by 2 gives me 0.35. Or in other words, I would get 35% sugar.

So you can see that I need to add more or less of one quantity over the other to get my 30%, and I think I'm going to be adding more of my 20% solution and less of my 50% solution because I'd like to bring this down from 35% to just 30%. So how am I going to then solve this equation? And the key thing here is to use my relationship between x and y. I know that x and y have to add up to equal 2, because I want only 2 quarts of my juice.

Well, I'm going to create an equivalent equation here. I am just subtracting x from both sides here, and I get that y equals 2 minus x. So I'm going to use this equation and substitute then 2 minus x in for y. So I'm just rewriting my equation, and whenever I see y, I'm going to write 2 minus x. And then I just know that x plus y equals 2, so my denominator here is 2. So now, I have this equation that has just one variable and I can solve this. So let's go ahead and solve this equation. First when I'm going to do is distribute 0.5 into 2 minus x. So 0.5 times 2 is one, and then 0.5 times this negative x is minus 0.5 x.

All right, so here's our equation again. Now, I'm going to multiply both sides by 2 so that I don't have a fraction. So I just have 0.2 x plus 1 minus 0.5 x equals 0.6. Remember, multiplying that by 2. Now, what I'm going to do is subtract 1 from both sides of this equation so I have all of my x terms on one side. So 0.2 x minus 0.5 x, and then when I subtract 1 from 0.6, I have negative 0.4.

Well, now let's combine our x terms. So 0.2 x minus 0.5 x is negative 0.3 x, and now I can divide both sides by negative 0.3. So I have x equals 1.33 when I round. So what is x equal again? Well remember, x is the number of quarts of my 20% solution. So returning back to our problem then, I know that I'm going to be adding 1 and 1/3 quarts of my 20% mixture. And because I need 2 quarts total, that means I'm just going to be adding 2/3 of a quart of my 50% mixture and that then is going to give me my 2 quarts of my 30% mixture.

So in our mixture problems, this is the equation that I like to utilize. To interpret this, this is the first quantity times the concentration of the first solution plus the second quantity times the second concentration, and we're going to divide that by our two quantities added together, and that's going to equal than our desired concentration. So that's how we can interpret that equation. So let's review mixture problems using weighted average. First, we talked about calculating simple average or mean. This is your adding up all of your data points and dividing by the number of data points. Weighted average has our data points being multiplied by its weight, so we consider some data points more important, so to speak, than others.

And when we're solving mixture problems, I like to use this equation here where we multiply the quantity times the concentration of one solution, and then we add to that the quantity times the concentration and the second solution, and we divide that by our total quantity, and that's going to equal our desired concentration. so Thanks for watching this tutorial on mixture problems and weighted average. Hope to see you next time.

Formulas to Know
Weighted Average

fraction numerator open parentheses C subscript 1 close parentheses open parentheses Q subscript 1 close parentheses plus open parentheses C subscript 2 close parentheses open parentheses Q subscript 2 close parentheses over denominator Q subscript 1 plus Q subscript 2 end fraction equals C subscript 3