Example 2 is another mixture problem.
There’s one little thing we have to make sure we catch in this problem that
wasn’t in the last problem, but it’s going to be pretty much the same process. I’m going to have two equations with two variables. In this case, I’m trying to
find how much of each type of solution and think about what a chemist has. A
chemist has different types of solutions. This particular chemist, she wants to
mix together one that has 15% alcohol in it with one that has 40% alcohol in it.
So she gets a solution that has 25% alcohol in it. It’s not as
straightforward as you might think, in terms of pouring those things in, so
we have to make sure that we use a system of equations to find the amount
that she should put together so that she ends up with a total of 30 ounces and
that the 10% mixes correctly with the 40% to give her 25% of solution of
alcohol. So, basically, the two things that she’s mixing together, that’s what we’re
trying to find. So X, in this case, is going to be the amount of 15% solution, and Y, in this case, is going to be the amount of 40% solution that she’s going to mix together. Okay? So the first equation just deals with the total amount of liquid that she’s going to have in this final mixture, or this final
solution. So if I take X + Y, that has to equal 30. So the total amount of the
15% stuff that she pours in, plus the total amount of the 40% solution, should
end up with 30 ounces of total solution. Then the second one puts
the weights. Okay, so now we talked about percent alcohol in the second equation.
All right, so we have 15% alcohol times the amount of solutions:
15%… plus 40% alcohol times the amount of that solution equals… now, here’s the
thing I have to be careful of, which wasn’t in the last problem.
I want my final solution to be 25%. Now 25% of what? Well, 25% of the total amount that’s in that
final solution is going to be alcohol, so the amount of alcohol that’s in the
final solution is going to be 25% of this 30 that I had from the previous
problem, or from the previous equation that I wrote here on the board. So now
these are my two equations. Take a look at that. You might want to put
something in your notes about when you have a solution like this, or a mixture, a
lot of times you have to take your final percentage times the amount that you
have of that solution. So let me go ahead and solve this one by substitution, since I did the last one by the combination/elimination method. So what I’m going to do is, I’m going to take this first equation and I’m going to rewrite it as
Y=30 – X. So now I’ve isolated Y, so every time I see a Y in the second equation, I’m going to substitute in (30 – X). So if I go over here, that will
give me 0.15x + 0.40(30-X), instead of 0.40(Y). Okay, so then that equals 25% of 30 is 7.5, I believe. So that’s the equation that I’m going to have there. So now I go about solving it. Okay, so I go ahead and distribute in this next step:
0.15x + (0.40*30… that gives me 12) – 0.40x is equal to that 7.5 that I have on the right-hand side. 25% of 30 is 7.5. So now I can combine these like terms and I’m going to get -0.25x or 25/100x or 25%. Okay it is equal to – if I bring this 12 over, if I subtract it, that’s going to give me a -4.5. Again, don’t worry about negative signs; we can divide out those negative signs, because in this last step I’m going to divide by 1/4 or divide by 0.25. Actually, a -0.25. So that’s the same thing as basically
multiplying by -4… 4.5(-4) or 4.5*4… is going to give me 18. Okay? So I’m not quite done there; I need to find a y-value. If X is 18, then 18 + y must give me 30, so Y must be 12.
Now I need to – whenever I do a real-life or an application problem, I need to put
units on my answer. So I need to go back and make sure that I understand exactly
what was asked. So the 18 is the amount of the 15% solution. So I need 18 ounces
of 15% alcohol solution, and then I need 12 ounces of the 40% alcohol solution. If I mix those together, I’m going to end up with 30 ounces total, and that solution is going to have 25% alcohol in that total amount of liquid.