>> Math is cool and you can do it.
>> We are going to work on a percent mixture problem using one variable.
So, here is a problem, how many pounds of dog food that is 50% rice, so that means half
of the dog food is pure rice, must be mixed with 400 pounds of dog food that is 80% rice
to make a dog food that is 75% rice.
So, the idea is we are taking two different concentrations of dog food and by concentrations
that means how much of it is pure rice and we are trying to make something
where the rice is somewhere in between.
So, here is the picture, we have got two dog foods here and we are going
to pour them together into this bigger vat, so we are going to imagine,
we are going to take some dog food that is 50% rice and some other dog food that is 80% rice,
and we are going to pour it into a large vat.
So the question is how much is in each vat?
So we look to see if it tells us how much is in any of these vats.
So, it does say here 400 pounds, that is the 80% rice,
so this 80% vat in that we know you have got 400 total pounds of dog food.
Now, that does not mean it is all rice, remember only 80% of that is rice.
And that is all what tells us right?
So the question is how many pounds of the 50% dog food and that is what we do not know.
So we will let that be our variable, let us call that x. We do not know how much is here.
But again, this is just the total amount of that package not just the rice.
So, if I pour it together, that is what I have got in here,
I have got x pounds plus 400 more pounds, so in this big vat I have x plus 400 pounds.
Now, let us now consider how much rice is in each of these vats.
Well, notice this is the 50% vat, writing that as a decimal is 0.50 or 0.5,
80% I am writing underneath that.
I am just writing the percent as decimals under each vat.
So, this is the amount of the actual bag, the amount of total food, but now I am going
to write out the amount of rice in each bag.
Well, 50% of the bag is rice of this first one, so that is 0.50 times x,
so that is how much rice is in this bag.
And how much rice is in this bag, 80% of the 400, so that is 0.80 times 400,
and how much is in this total vat then, I know I want to make it so it is only 75%,
so instead of being 80% it is only 75% which is more than the 50%.
You can guess there is going to be more of the 80% right, because it is only going down from 80
to 75, so this could be 0.75 times x plus 400.
Okay, alright.
So we have this.
That is how much rice is in each, right, but since we are pouring the pure rice that is
in each vat together, the pure rice in the final should be the same.
So, if I add the pure rice from the first bag and you imagine how much pure rice is
in the second bag, you should end up with exactly the same amount of rice
in the third bag, and this is your equation.
So the equation comes from the amount of pure rice that is in each bag,
so imagine if you were going to look at the picture, let us say half of this is rice,
so let us get a little picture of that.
So, I am going to get a highlighter, sorry.
Alright, so this is like about 50% rice, let us say, let us say about half
of it is rice you can imagine, then over here it is like a lot more, it is like about 80%
so you get a picture of the yellow part is how much is pure rice, right,
out of the whole container, and the answer is about 75% rice.
You could sort of see a picture.
That is where we are getting our equation from is how much pure rice is in each container,
so you can not all of a sudden have more rice or less rice in the final container
than they were in the original two.
Alright, so let us just think what we have done so far.
We drew a picture what is going on and in that is where I am really defining x. So,
if you want you could write somewhere else that x is how much dog food you are using of 50%
or you could just draw a picture and it is clear that is what x stands for and we fill
in what we know in these three places, there will be some unknowns
and to get the equation is the actual amount of rice.
So the next step would be to solve this equation.
Now, you could solve this equation by working
with the decimals throughout the problem using a calculator.
I like to multiple both sides of the equation by a 100 to eliminate all of the decimals
and since 0.75 has two places after the decimal point, I am going to multiply everything by 100.
So, we are going to multiply both sides by 100, which means I am multiplying 100
on the left side and 100 on the right side.
Now, when you are multiplying 100 times 0.75 that will take care of that decimal,
but the x plus 400 does not also get multiplied.
Alright, so what we have here is 100 times 0.50 will just now be 50x
and that will just be 80 times 400 and that will be 75 times x plus 400.
Now, when you write the original equation you do need to have the decimal points
it really was percents, the only reason I could write it this way is because it is
in equation and I multiply both sides by 100.
Alright, so this gives me 50x plus 80 times 400, well that is 8 times 4 which is 32
and then the three 0's, and then we distribute the 75, so 75 times x is 75x, and 75 times 4,
let us see, 75 times 2 is 150 times 2 again is 300, with the extra two 0's, okay.
So, I have got each side simplified
and now we could get the variables on either side of the equation.
I am going to subtract 50x from both sides so that the x term will be positive,
so if I subtract 50x from both sides I have got the x terms on the right side of the equation
which will make a positive right.
And that means we do not want the 30,000 on this side, so I am going have to subtract 30,000
at same time, if you want and that will give me 2,000.
Alright, so the last thing to do is to divide both sides by 25.
[ Pause ]
>> Alright.
Let's see while 25 goes in the 208 that is going to be 80, so x is going to be 80,
so that is what I have got so far.
Now, you have to remember that that might not be the question they are asking for.
We have to go back to the original equation and I want to just point out,
this original problem you could have just done the distributive property using all the decimal
points here and done everything with a calculator and you should still get x equals 80.
I just did it without a calculator by multiplying by 100.
So, our next step is go back to the problem, the original problem, and then we are going
to draw a picture and see if this all makes sense again.
So here was our original problem, how many pounds of dog food
that is 50% rice must be mixed with 400 pounds of dog food that is 80% rice
to make the dog food to 75% rice, and this is how we set up our variable.
We said x was in the 50%, 400 was in the 80%, so now you know what x is.
Now, we have to check our work, x was 80 and in our original picture we had put
that with a 50% vat, so we know what that is.
So, we have got 80 pounds and 400 pounds we are mixing together
that means there must be 480 pounds over here.
Now, let us see how much rice is in each bag, so since 50% of 80 so 50%
of 80 is half of 80 or 40 pounds.
In this bag it is 80% of 400, so 0.8 times 400, that is 320 and in the mixture we want 75%,
0.75 times 480, so that ends up being 360.
So, the question is if you have this particular mix, 80 pounds, 400 pounds
and you mix it together, would you have the same amount of rice in the end.
So, the question is, is 400 plus 320 really the same thing as 360 which is what is in there?
Yes it is, so we answered the question.
The question was how many pounds of dog food that is 50% rice?
There it is we look at the picture, it is 80 pounds.
So remember, write your answer in words, x is not necessarily always going to be the answer.
>> I hope you learn something, remember math is cool.