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Math is cool and you can do it.
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Okay, we're going to work on a percent mixture problem using one variable.
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So here's a problem. How many pounds of dog food? That is 50% rice.
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So that means half of the dog food is pure rice.
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Must be mixed with 400 pounds of dog food that is 80% rice
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to make a dog food that is 75% rice.
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So the idea is we're taking two different concentrations of dog food.
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And by concentrations that means how much of it is pure rice.
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And we're trying to make something where the rice is somewhere in between.
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So here's the picture.
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We've got two dog foods here.
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And we're going to pour them together into this bigger bin.
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Right? So we're going to imagine we're going to take some dog food that is 50% rice.
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And some other dog food that is 80% rice.
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And we're going to pour it into a large fat.
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Okay, so the question is how much is in each fat?
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So we look to see if it tells us how much is in any of these fats.
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So it does say here 400 pounds that is the 80% rice.
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So this 80% fat in that we know we've got 400 total pounds of dog food.
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Now that does not mean it's all rice.
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Remember only 80% of that is rice.
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And that's all it tells us, right?
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So the question is how many pounds of the 50% dog food?
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And that's what we don't know.
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So we'll let that be our variable.
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Let's call that x. We don't know how much is here.
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But again, this is just a total amount of that package, not just the rice.
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So if I pour it together, that's what I've got in here.
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Plus 400 more pounds.
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So in this big fat, I have x plus 400 pounds.
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Now, let's now consider how much corn, I'm sorry, how much rice is in each of these fats.
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Well, notice this is the 50% fat.
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Writing that as a decimal is 0.50 or 0.5.
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80% I'm writing underneath that.
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I'm just writing the percent as decimals under each fat.
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So this is the amount of the actual bag, the amount of total food.
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But now I'm going to write out the amount of rice in each bag.
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Well, 50% of the bag is rice of this first one.
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So that's 0.50 times x.
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So that's how much rice is in this bag.
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Now, how much rice is in this bag, 80% of the 400?
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So that's 0.80 times 400.
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And how much is in this total fat then?
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I know I want to make it, so it's only 75%.
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So instead of being 80%, it's only 75%, which is more than the 50%.
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You can guess there's going to be more of the 80% right?
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Because it's going to be going down from 80 to 75%.
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So this would be 0.75 times x plus 400.
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Okay.
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All right, so what we have is, that's how much rice is in each, right?
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But since we're pouring the pure rice that's in each fat together,
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the pure rice and the final should be the same.
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So if I add the pure rice in the first bag,
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and you imagine how much pure rice is in the second bag,
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you should end up with exactly the same amount of rice in the third bag.
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And this is your equation.
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So the equation comes from the amount of pure rice that is in each bag.
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So imagine if you were going to look at the picture,
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let's say half of this is rice.
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Let's get a little picture of that.
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So I'm going to get, I'm, highlighter, sorry.
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All right, so this is like about 50% rice, let's say,
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let's say about half of it is rice.
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You can imagine, then over here it's like a lot more.
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It's like about 80%.
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So you can get a picture if the yellow part is how much is pure rice, right?
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Out of the whole container.
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And then yeah, it's about 75% rice.
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You could sort of see a picture.
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That's what we're getting our equation from.
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It's how much pure rice is in each container.
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So you can't all of a sudden have more rice or less rice in the final container
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than there were in the original two.
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All right, so let's just think what we've done so far.
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We drew a picture of what's going on.
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And in that is where I'm really defining X.
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So if you want, you could write somewhere else that X is how much dog food
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you're using of 50% or you could just draw a picture and it's clear.
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That's what X stands for.
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And we fill in what we know in these three places.
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There will be some unknowns.
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And then we, to get the equation, it's the actual amount of rice.
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So the next step would be to solve this equation.
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You know, you can solve this equation by working with the decimals
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throughout the problem using a calculator.
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I like to multiply both sides of the equation by 100
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to eliminate all of the decimals.
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And since 0.75 has two places after the decimal point,
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I'm going to multiply everything by 100.
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So I'm going to multiply both sides by 100, which means I'm multiplying 100
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on the left side and 100 on the right side.
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Now when you're multiplying 100 times the point 75,
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that'll take care of that decimal, but the X plus 400 does not also get multiplied.
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All right, so what we have here is 100 times 0.50.
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We'll just now be 50X.
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And this will just be 80 times 400.
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And that'll be 75 times X plus 400.
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Now when you write the original equation, you do need to have the decimal points.
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It really was percents.
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The only reason I could write it this way is because it's an equation.
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And I multiply both sides by 100.
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All right, so this gives me 50X plus 80 times 400.
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Well, that's 8 times 4, which is 32.
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And then the 3 zeros.
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And then we distribute the 75.
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So 75 times X is 75X.
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And 75 times 4.
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Well, let's see, 75 times 2 is 150 times 2 again is 300 with the extra 2 zeros.
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Okay, so I've got each side simplified.
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And now we can put the variables on either side of the equation.
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I'm going to subtract 50X from both sides so that the X term will be positive.
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So if I subtract 50X from both sides, I've got the X terms on the right side of the equation,
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which will make a positive, right?
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And that means we don't want the 30,000 on this side.
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So I'm going to have to subtract 30,000 at the same time if you want.
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And that will give me 2,000.
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All right, so the last thing to do is to divide both sides by 25.
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All right.
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You see, well, 25 goes into 208.
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So that's going to be 80.
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So X is going to be 80.
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All right, so that's what I've got so far.
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All right, now you have to remember that that might not be the question they're asking for.
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We have to go back to the original equation.
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And I want to just point out this original problem.
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You could have just done the distributive property using all the decimal points here,
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and then everything with a calculator, and you should still get X equals 80.
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I just did it without a calculator by multiplying by 100.
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So our next step is go back to the problem, the original problem,
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and then we're going to draw a picture and see if this all makes sense again.
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All right, so here is our original problem.
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How many pounds of dog food that's 50% rice must be mixed with 400 pounds of dog food that's 80% rice
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to make a dog food the 75% rice since how we set up our variable.
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We said X was in the 50%, 400 was in the 80%.
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So now we know what X is.
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Now we have to check our work.
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X was 80, and in our original picture we had put that with a 50% vet,
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so we know what that is.
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So we've got 80 pounds of 400 pounds from mixing together.
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That means there must be 480 pounds over here.
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Now let's see how much corn is in each, sorry, rice is in each bag.
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So it says 50% 80.
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So 50% 80 is half 80 or 40 pounds.
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In this bag it's 80% of 400, so 0.8 times 400, that's 320.
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And in the mixture we want 75%, 0.75 times 480, so that ends up being 360.
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So the question is if you have this particular mix, 80 pounds, 400 pounds,
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and you mix it together, would you have the same amount of rice in the end?
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So the question is, is 400 plus 320 really the same thing as 360, which is what's in there?
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Yes it is.
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So we answer the question. The question is, was how many pounds of dog with this 50% rice?
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There it is.
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We look at the picture, it's 80 pounds.
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So remember to write your answer in words, X isn't necessarily always going to be the answer.
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I hope you learned something, remember math is cool.