The ratio, by weight, of the four ingredients A, B, C, and D of a cert

First, make a quick table to represent the ratios in the original mixture:

A B C D
4 7 8 12

One way forward is to make the A:C change, holding A:B and A:D constant. (We know that A:D decreases, but we don’t know by how much, so for now, pretend that the ratio stays constant.)

To quadruple the ratio A:C, we can either multiply A by 4 or divide C by 4. Since we want to leave A:B and A:D constant, it’s more efficient to divide C by 4. So if A:D weren’t changing, the new mixture would have these ratios:

A B C D
4 7 2 12

Since A:D decreases, let’s add an unspecified amount to D. (We don’t want to mess with the A side of the ratio, since A is involved in other ratios.) Call that amount x.

A B C D
4 7 2 12 + x

Now we know that B will constitute 20% of the whole, so the ratio of B to the whole in the final mixture will be 20 to 100, or 1:5. If we look at the table, the ratio of B to the whole is 7 to 25 + x. We can equate these proportions:

1/5 = 7/(25 + x)
25 + x = 35
x = 10

So now we know that the final mixture has these proportions:

A B C D
4 7 2 22

The new ratio of A to D is 4:22, or 2:11. As a fraction, this ratio is 2/11.

The original ratio of A to D is 4:12, or 1:3. As a fraction, this ratio is 1/3.

Finally, we are asked this: if 1/3 is decreased to 2/11, what is the percent decrease?

A fast way to compute this number is first to figure out the factor that you multiply 1/3 by to get 2/11. Call that factor y.

(1/3)y = 2/11
y = 6/11

So 2/11 is 6/11 of 1/3. As a percent, 6/11 is approximately 55% (as a decimal, 6/11 = 0.5454…). If the new number (2/11) is 55% of the old number (1/3), then the percent decrease is 100% – 55%, or 45%.

The correct answer is D.