solve ratio word problem without algebra

Question

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.

Answer 1

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.

Answer 2

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 $2/3,4/3$.