This problem can be a trick problem as adding the two ratios just won't do the job here! For example, simply adding a can of A and a can of B won't get you 1:5! (Even though (1+1):(3+7) = 2:10=1:5).
So, I had to go deeper. For the first question, I found out that the two ratios, if each can had 1 litre, we would get 2/8 and 6/8 (1/4 and 3/4) for A and 1/8 and 7/8 for B. To make this easier, I just assumed that each can was 8 litres! Then we would get whole number values for the volume of each colour paint.
Next, I thought: What can I do with this new property that I found? The answer was an equation.
The equation that I came up with was largely based on what was required as the end result. The end result was a ratio between red and white paints. This ratio can be expressed as (2A+B)/(6A+7B). The numerator is the total volume of red paint and the denominator is the total volume of white paint, with A and B as the number of 8L cans of A and B respectively.
The last step of making an equation is putting this equal to the ratio of the desired paint.
Now we can solve these equations and get a result similar to 3A=4B (this is an example). We can see that the smallest possible pair of solutions that is a whole number is A=4, B=3.
Using this logic, you can quickly find the needed numbers of each can.
You can use a similar logic for the second set of questions, except the can size will be 10 instead of 8.
Finally, is it always possible to combine two paints made up in the ratios 1:x and 1:y and turn them into paint made up in the ratio 1:z (where x<z<y)?
Yes, of course because you can always do the steps that I have done above to any ratio of paint in the form of 1:w.