We start with two paint bottles. The first bucket with the ratio of red to white paint being 1:3 and the second one being 1:7.
To find how many buckets of each you need, you first call the number of ratio 1:3 bucket 'x', and the number of ratio 1:7 bucket ' y'.
Then we find out the ratio of red to white paint on the first bucket using fractions (you don't have to but it is easier to understand). We find that it is 1/4 : 3/4. From here, because our main problem is to get how many buckets we use to make a paint with the ratio of red to white of 1:k we substitute an 'x' in this ratio and we get (1/4)x : (3/4)x.
Then we do the same thing for the second paint bucket using 'y' this time, and we get (1/8)y : (7/8)y.
In order to solve for how many buckets of each we mix to make the paint with ratio 1:k we create the ratio:
(1/4)x + (1/8)y : (3/4)x + (7/8)y
This is the equation as on the left side we have the ratio of all the red paint and on the right, we have all the ratio for the white paint. To find for the ratio 1 : k we set it as the ratio above is equal to 1 : k. For example, if k was 4, we would create the equation:
(1/4)x + (1/8)y : (3/4)x + (7/8)y = 1 : 4
From here we can cross multiply, and we get:
(3/4)x + (7/8)y = 4 ((1/4)x + (1/8)y)
If we solve this equation we get how many buckets of each we need.
(3/4)x + (7/8)y = 4 ((1/4)x + (1/8)y)
(3/4)x + (7/8)y = x + (1/2)y
(3/8)y = (1/4)x
3y = 2x
From this equation we could get any number, however, because the question asks for the lowest number of 'x' and 'y', we know that 'x' is 3 and 'y' is 2.
For any value of k, we use the added ratio, (1/4)x + (1/8)y : (3/4)x + (7/8)y, to solve it.