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Four gallons of yellow paint plus two gallons of red paint make orange paint. I assume this makes six gallons. So the ratio is 4:2, or 2:1.

Question: how many gallons of yellow paint, and how many gallons or red paint, to make two gallons of orange paint?

2y + r = 2
2y + y = 2
3y = 2
y = 2/3

or

4y + 2r = 6
(4y + 2r)/3 = 2

so I get 4/3 and 2/3.

However, in this section of the text book, I'm not sure that it's "allowed" to do any of that. Is it possible to solve this just with cross-multiplying a ratio?

Their examples setup a ratio with an unknown n, cross multiply and solve for n. I don't see how to solve this word problem with that technique.

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  • $\begingroup$ Let us "guess" that it is $4$ of yellow and $2$ of red. Oops, wrong guess, we got $6$ gallons. Scale by the factor $\frac{2}{6}$. $\endgroup$ Commented Feb 10, 2014 at 5:43
  • $\begingroup$ huh? no, the 4:2 is given in the question. I assume that 4 gallons of yellow plus two gallons red equals six gallons. Not all chemicals work that way, sometimes 2 gallons of x plus 4 gallons of y only give 5 gallons. I don't know what you're saying about a guess of 6....? It's not a guess, it's an assumption not stated in the question. Yes, scale by 1/3....but how? There's no algebra in this section. $\endgroup$
    – Thufir
    Commented Feb 10, 2014 at 5:48
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    $\begingroup$ Yes, the $4:2$ is given in the question, and yes, alcohol and water don't behave this way, but we have been asked to assume paint does. What I am saying is that since you know that $4$ and $2$ give $6$, to get $2$ we must scale by $\frac{2}{6}$, giving $4\cdot\frac{2}{6}$, $2\cdot\frac{2}{6}$. $\endgroup$ Commented Feb 10, 2014 at 5:52
  • $\begingroup$ ok, I see what you're saying. interesting. $\endgroup$
    – Thufir
    Commented Feb 10, 2014 at 5:55
  • $\begingroup$ Claude Leibovici has given a clearer exposition. $\endgroup$ Commented Feb 10, 2014 at 5:59

2 Answers 2

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Hint

Take the problem in the other way.

You are told that $6$ gallons of orange paint are made mixing $4$ gallons of yellow paint and $2$ gallons of red paint.

Divide these numbers by 6 in order to come back to one gallon of orange paint. Then, one gallon of orange paint is made mixing $\frac{4}{6}=\frac{2}{3}$ gallons of yellow paint and $\frac{2}{6}=\frac{1}{3}$ gallons of red paint.

Now, multiply by $n$ which is the number of gallons of orange paint you want to make.

So, making $n$ gallons of orange paint require mixing $\frac{2n}{3}$ gallons of yellow paint and $\frac{n}{3}$ gallons of red paint.

Now, you want $n=2$; then ....

I am sure that you can take from here.

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  • $\begingroup$ ok, this is interesting, thanks. $\endgroup$
    – Thufir
    Commented Feb 10, 2014 at 5:53
  • $\begingroup$ You are welcome. Thinking simple helps quite often. If you like my answer, you can accept it. Cheers. $\endgroup$ Commented Feb 10, 2014 at 5:57
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The ratio is $2:1$ and so their values are $2x,x$ and their sum is $3x$. to get 2 gallons we put $3x=2\implies x=2/3$. thus the required values are $2/3,4/3$.

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  • $\begingroup$ exactly, but that seems to use algebra, or the beginnings of algebra. In this section they don't seem to have substitution. $\endgroup$
    – Thufir
    Commented Feb 10, 2014 at 5:50

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