Int Algebra: Solving a Mixture Problem

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08/13/2009

~ Dustin3539

Seriously...Algebra has never been this easy to understand. Thanks for the help and keep the great lessons coming!

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Equations and Inequalities
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Solving a Mixture Problem Page [1 of 2]
Okay, as you can see, these real world word problems can get pretty hairy. Let’s take a look at another one together
and see if we can make progress on this.
This one that is the following: it’s about mixture. This is called a mixture problem. Let me try to give you a little
context to it. Suppose that you wanted to make some fine jewelry, have some fine jewelry custom made for your very
significant other and you’d like it to be silver, but the excuse that you're going to give is that silver is a very malleable
kind of metal, so if you gave your special person one of these things, it might bend and break. And so you're going to
mix in some alloy, some stronger metal to actually make the thing more rigid. Of course, that’s the excuse you give.
The truth is we know that you're just really cheap and so you don’t want to actually spend a lot of money on silver, you
want to put in a lot of junk. So the question is how much silver to junk should you put in if you want a certain thing.
So let’s suppose that, for making some fine thing like this, like a little doggy chain, you need a certain ratio. Let’s take
a look at the question and see what we have to do.
So in this question we want to know how many grams – things are measured in grams – how many grams of pure
silver, that’s the expensive stuff, a silversmith mixes together with a 45% silver alloy to produce 200 grams of 50%
alloy. So what that’s saying is the following: you’ve got some pure silver and then you’ve got some diluted silver, and
you have 45% diluted silver. Well, of course, if you mix them together, since this is pure silver here and this is only
45%, as you mix them together, in some sense, the percentage of silver will be higher than 45, but it won’t be pure
anymore, because it’ll be contaminated. So the question is we want to get 200 grams that actually is 50% silver and
50% junk.
So let’s take a look at this and think about it now together. So here we have a whole bunch of really expensive silver
stuff. This is the stuff you don’t want to spend money on, and then we’ve got all the junky stuff. Here’s the junky stuff
and here’s the silver stuff. So the question is I want to take some of the silver stuff and melt it down and mix it up with
the junky stuff in some way. And then when I put it all together, I have this mish-mash of 50% alloy. So that’s what I
want to do. How do I mix them?
So let’s see if we can make some progress on this. So the question is how much silver do we need? So let me call
the amount of silver I'm going to need s. So this is an unknown that I'm introducing. It’s there, but it’s not written
mathematically. So s is the amount of silver. Okay, if s is the amount of silver, can we actually figure out how much
of the alloy stuff we’re going to add? Now, the answer is yes and the answer is hidden in the question. If we look at it
again, we’ll see that, at the end of the day, well, we’re going to want a 50% alloy, that’s not the important thing. But
notice we’re going to want 200 grams. So I'm going to be adding some pure silver and I'm going to be adding some of
this diluted stuff, but, at the end of the day, the total amount is going to be 200 grams. So that 200 grams, where does
it come from? There will be some silver. In fact, how much silver? Well, we already said there will be s silver, and so
the rest is going to be that fake stuff. So if there’s 200 total and this is s, what must this amount be? Well, it must be
200 – s. So, in fact, the amount of the cheapo stuff is going to be 200 – s, and that equals the junky stuff. If you add
those two amounts together, we get the 200 total.
Okay, so notice that I’m using the total number of weight to figure out how much junk I have compared to how much
silver. But now what I’ve got to do is – I can’t solve that for silver yet, because I don’t have a relationship with those
things together yet. But what do I know? I know that when I’m all done with that big blob of 200 stuff, how much of it
should be pure silver? How much of that 200 grams should be pure silver? Look at the question again. It says that
this thing should be 50% alloy, which means also 50% good stuff. So if the thing is 200 grams in total, how much pure
silver is there? 100, right? There’s 200 total, 50% of it is junk, that means that 50% of it is the good stuff, so we know
there’s 100 grams of pure silver. So we know – now, look how I'm using all that information together. See, this is a
little tricky. You’ve got to think about it. We know there’s 100 grams at the end of the day of good stuff in the final
thing.
Okay, well if we know there’s 100 grams of good stuff, all we have to do now is figure out how much contribution of
good stuff we’re getting from the good stuff and then the junky stuff. So let’s think about that. I’ve got this 100 grams
of pure silver in our final mixture. How much of that came from the silver? Well, the silver contributed all of s grams,
because this is pure silver. We added that in, that’s just pure silver. So, in fact, that contributed all of its stuff. But, in
Equations and Inequalities
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Solving a Mixture Problem Page [2 of 2]
fact, we have 200 – s of junk, but not all of that is pure silver. That’s the whole point. In fact, only a fraction of it is
pure silver. Only 45% of it is silver. So that means that how much does it contribute to this 100 grams of final mixture
of pure silver? Well, 45% of this. And what’s 45% of something? It’s just .45 times the amount.
So let me recap what I just did here. I know I have 200 total, but out of that 200 only 50%, or 100 of it, is pure silver.
And now I want to see how that pure silver got there. Well, it got there because I added in s amount, s grams of just
pure silver, and then, out of all this junk, it actually contributed 45% of the junk contributed to our silver. So pure
silver, and this is the pure silver from the pure silver. This is the pure silver from the junk contribution, and that total
equals 100.
Okay, the hard part is behind us. Now we just have to solve the dumb thing. You can see that the actual mass stuff is
not that hard. The real hard stuff is converting from the words to here. And I’ll tell you something, in my opinion,
that’s really what the math is, going from the words to here. This is the math. The rest of it now is just routine stuff.
So again, think about how I did it. I knew how much I had in total. That gave me the ability to say how much silver I
have compared to how much junk I have. And then, knowing how much the actual amount of pure silver there was
going to be in the final answer, 50% or, in this case, 100 grams, I can figure out the contribution of each of those
things and now I can solve.
So let’s do that. s plus, and I'm going to distribute the .45. If I distribute the .45 throughout here, I'm going to see that
this is going to be – how can I write this? Well, I can write this as
100
45 , that’s what .45 means, times 200 and then
minus .45s. Notice, I'm distributing. Again, don’t make any kind of classic mistakes now. This is not a good time to
do it. Everyone gets hit with that .45 and that equals 100.
Okay, well let’s see, I have s and I subtract off .45s. And so I think that leaves me with .55s. And then here I have
some cancellations. Those zeros can cancel out and I'm just left with 45 × 2. 45 × 2 = 90, and I have 100 on this
side. If I bring the 90 over, I'm subtracting 90, which would give me 10, so this would equal 10. And so what do I
see? I can just divide by sides by .55 s, and I would see that s =
.55
10 . And if you work that out, if you just divide that,
you'll see 18 and
11
2 grams. The units here are grams.
So the question is how much silver should we add to this thing to make this beautiful dog collar? Well it turns out that
the answer we need 18 and
11
2 grams of pure silver, therefore the rest has to the junk. So if I put it all together, I get
exactly 50%. By the way, does this answer seem reasonable? Well I think it sort of does in the following way. This is
not going to be a mathematical argument, but think of it this way: look, we have this 45% junk stuff at the beginning,
and we only want to boot it up to 50%. So if we just took the 45% stuff and used no pure silver, we’re pretty close to
the final answer we want, because it’s 45% pure stuff and I want 50%. So I just have to add a little teeny bit to top off
that 45% and make it up to just 50%. So we shouldn’t be adding an awful lot, if you think about it, because the junky
stuff is so close to the right answer anyway.
It’s good to think about it and see if you can reason through and make sure that the answer is sensible. For example,
if we got an answer, let’s say we got an answer of 300 grams. That would be a very bad answer. Do you see why?
Because the total is supposed to be 200. How can I put in 300 pure silver – it just doesn’t make sense. So always
think about if the answer at least makes sense. This makes sense and hopefully it’s also correct.
So you can make your cheapo dog collar, give it to your special friend and that person can wear it wherever they wear
these things and life is happy. Enjoy.