Let the number of scoops of paint needed be R and W for red and white paint.
The total number of scoops when R and W are added will be 5+R+W. The amounts of paint when the larger batch is made up will be 2+W and 3+R. The fraction of the total batch has to be (2+W)/(5+R+W)=3/5 and (3+R)/(5+R+W)=2/5.
So we have two equations: 10+5W=15+3R+3W; 2W-3R=5; and 15+5R=10+2R+2W; 2W-3R=5, which is the same equation. So W=(5+3R)/2, which means that 5+3R must be an even number, therefore R must be odd. The smallest odd number is 1 so W=4. That's 4 scoops of white and 1 scoop of red.
(If R=3, W=7 so 10 scoops would be added, totalling 15 scoops, containing 6 red and 9 white. The relative amounts are 6/15=2/5 and 9/15=3/5. And so on. The question amounts to equivalents of the fractions 2/5 and 3/5 by multiplying top and bottom by the same integer and making up the deficiencies in the scoops. 20/50 requires 17 scoops of red and 30/50 requires 28 scoops of white, for example. The whole of the paint can be used up in this way. The white paint uses 4, 7, 10, 13, etc., scoops and will be used up first (if the red and white tins have the same size), so when there are fewer than 3 scoops left of the white paint, that will be the limit.)
CHECK: The new batch has 5 extra scoops making it 10 scoops. Of these 4 are red and 6 are white, so that's 4/10=2/5 and 6/10=3/5, and the ratio of red to white is 2:3.