You need to learn better how to simplify fractions. "Canceling" doesn't just mean "cross the same thing off in two places and they disappear"; it means dividing the (entire) numerator and denominator of a fraction by the same thing, or equivalently removing a common factor, to get an equivalent fraction.
Here is the correct work for dividing V1+V2 by V2 and simplifying, which I do by distributing the division (dividing each term separately):
\(\displaystyle \dfrac{V_1+V_2}{V_2} = \dfrac{V_1}{V_2} + \dfrac{V_2}{V_2} = \dfrac{V_1}{V_2} + 1\)
Here I simplified the second fraction by dividing the numerator and denominator by V2 to get 1; I couldn't do that in the original form because V2 was not a factor of the entire numerator.
As for your work on the bigger problem of which this is a tiny part, you seem to be blindly combining things without a goal. The reason for using V1 = (D2V2)/(4D1) is to eliminate a variable so you can simplify; if you also use V2 = (4D1V1)/D2, then you still have both variables, and all you've done is to make things more complicated.
I'm not sure the method you are being shown elsewhere is correct, but if you want to follow it, try to think about why each step is being done so you can do it rationally. My goal at the moment is just to help you handle canceling correctly.
But perhaps you need to quote for us the entire original problem, so we can check whether this person is misleading you. One trouble I see is that he told you volumes don't add up, but then he used V1 + V2 anyway!
OK, I'm quoting everything here.
''Let me then give an alternative solution, if I may, that would be the preferred way of computing this quantity: D3=(D1V1+D2V2)/(V1+V2). In addition M2=4M1 so that D2V2=4D1V1.Numerator and denominator of the first equation can be divided by V2, and this can be readily solved for D1, (by also using the second equation). It is difficult sometimes to assess exactly what level the student is at, but this second solution would be a much improved approach, as its correct that these are normally specified 20% vs. 80% by mass.
Suggestion for the OP
@Richie Smash : See if you can follow what I did in post 17, and see if you can come up with the answer for D1 that you get from those two equations. (Do not use V1/(V1+V2)=.2 and V2/(V1+V2)=.8, because we have determined that that is not what a 20%- 80% composition refers to). If my arithmetic is correct, I get a slightly different answer than 7.2 gm/cm^3. I'd be interested in seeing if you get the same thing I did.
You have two equations and two unknowns: D1 and r=V1/V2. It may look like 2 equations and 3 unknowns: D1, V1 and V2, but the ratio of V1 to V2 shows up in both equations, so it simplifies. It takes a little more algebra to solve it this way, than with the previous assumption of 20-80 by volume, but the previous assumption is really a false assumption that lacks validity. The 20-80 by mass is really the way this problem should be worked.''
Then I did this
D3 = (D1V1 +D2V2)/(V1+V2)
D3 = (D1V1 +4 D1 V1)/(V1+V2)
D3 = (D1+4D1V1)/V2
D3 = D1
And he said resuming quote.
''Your 3rd line is incorrect. Let me show you what you should get when you divide numerator and denoninator by V2: D3={5D1 (V1V2)}/{(V1V2)+1}. Also hang on to the second equation D2V2=4D1V1, and write it as r=(V1V2)=D2/(4D1) and substitute it into the equation that I just wrote out for you. You got to work at getting the algebra correct in these problems. Much of the introductory physics is algebra.
Read post 26 again please. I may have edited it since you last looked at it. Simply substitute r=V1/V2=D2/4D1 back into the first equation: D3=(5D1r)/(r+1). The result that you get will be a somewhat clumsy expression which can then be simplified somewhat, and solved for D1. And the algebra in this one is non-trivial as algebra goes, but you need to stick with it, if you want to get the correct answer.
The first thing you need is to take the equation that is of the form D3=A/B and divide numerator and denominator by V2: D3=(A/V2)/(B/V2). Then process each part A/V2 and B/V2 separately. You should also have recognized that D1V1+4D1V1=5D1V1. Doing the complete algebra: (1)(D1V1)+(4)(D1V1)=(4+1)(D1V1)=5D1V1 . For the above A=5D1V1, and B=V1+V2. It should be a simple matter to divide A by V2, and B by V2, and then substitute in for r=V1/V2=D2/(4D1)''
END QUOTE
I hope this gives more insight, because I've been struggling.
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