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Applied Linear Equations: Mixture Problem - Problem 1

Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

Another applied linear equation problem, and this is again dealing with mixtures. So, in this one, a nut farmer mixes some peanuts that sell for 250 a pound, with some almonds, that sell for $5 a pound, to make a 12 pound mixture, worth $3 a pound, and it's asking for, how many pounds of peanuts were in the mixture?

So, the first step I always do in these is, to go and take our word problem, and make a diagram out of it. So, we have a bunch of peanuts that are 250 a pound, so this is 250 a pound, and we have a certain amount, but we don't know how much that is. So we can call it x. We're adding in almonds, which are $5 a pound and again, we don't know how much this is, so we're actually going to leave this blank, for right now, we'll come back to it in a second, and we're adding these together and our result is $3 a pound and 12 pounds.

So, we have a certain amount of peanuts, a certain amount of almonds and in total we have 12 pounds. If we had, just say, one pound of peanuts, we know our total has to be 12. So, we know that this has to be 11 pounds of almonds. If this was 4 pounds, we know our total also has to 12, so this has to be 8. So, there's always that difference where these two have to add up to 12. So if we actually take x away from 12, that will result with the amount of almonds. So this is actually 12 minus x. You can always check to make sure if this was 3, the 12 minus 3, 9; 3 and 9 would equal 12. So no matter what xs, these two are always going to add up to 12 pounds of mixture.

So we have a diagram. The second step is going to be turning this diagram into a linear equation. If we have one pound of almonds, this whole bag would sell for 250 just the one pound times the amount it's selling for. If this was two pounds, it'll just be 2 times 250, so basically the amount that's worth is the amount you have times the cost that it goes for. So this is just going to be 250 times x is the amount that your peanuts are worth.

By that same logic; cost times weight is the amount that your almonds are worth and you have $3 a pound, 12pounds this is just going to be 3 times 12 $36 worth of nuts in our final bag. We've taken our diagram changed it into a linear equation.

So for this one, we could either multiply by 10 or 100 or just leave at end, for this one I'm actually just going to leave in a decimal to show you it works either way. So this decimal doesn't actually do anything 2.50 is the same thing as 2.5 so it's 2.5x, distributing our 5 in 5 times 12 is 60 so we're left with 5 minus 5x and then this is equal to 3 times 12, 36.

Combining like terms 2 and a half minus 5, this is just going to give us -2.5x and then plus 60 is equal to 36, subtract that 60 over, 36 minus 60 is -24 and then to solve for x just divide by -2 and a half. 24 divided by 2.5, x is equal to 9.6.

So we only need to make sure we're giving what the question is asking for and this question was asking for how many pounds of peanuts were in the mixture, amount of peanuts we had was x, so that corresponds to this x right here so just make sure we have our units, we have 9.6 pounds of peanuts. As for almonds, just plug in 9.6 back into here and solve it out.

So we've taken our word problem, made a diagram, taken it into an equation and solved it out.

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