The ratio, by weight, of the four ingredients A, B, C, and D of a certain mixture is 4 : 7 : 8 : 12. The mixture will be changed so that the ratio of A to C is quadrupled and the ratio of A to D is decreased. The ratio of A to B will be held constant. If B will constitute 20% of the weight of the new mixture, by approximately what percent will the ratio of A to D be decreased?

Question: The ratio, by weight, of the four ingredients A, B, C, and D of a certain mixture is 4 : 7 : 8 : 12. The mixture will be changed so that the ratio of A to C is quadrupled and the ratio of A to D is decreased. The ratio of A to B will be held constant. If B will constitute 20% of the weight of the new mixture, by approximately what percent will the ratio of A to D be decreased?

Solution:
Lets say our thing has the weight x(4 + 7 + 8 + 12) = 31x.
When we change mixture our weight changes to x(4 + 7 + 2 + 12y)
=13x + 12xy

We also know that 7x = 0.2(13x + 12xy)
Reduce that last equation by x and lets just solve it for y : y = {7 - (0.2 × 13)}/2.4
= 11/6
So the original ratio was 4/12 = 1/3
The new ratio is (1 × 6)/(3 × 11) = 2/11
The resulting percentages are: {(1/3) - (2/11)}/(1/3) = 1 - (6/11)
= 5/11
= 0.4545
≈ 45%