The treatment of a certain viral disease requires a combination dose of drugs D1 and D2. Each unit of D1 contains 1 milligram of factor X and 2 milligrams of factor Y, and each unit of D2 contains 2 milligrams of factor X and 3 milligrams of factor Y. If the most effective treatment requires 13 milligrams of factor X and 22 milligrams of factor Y, how many units of D1 and D2 should be administered to the patient?
One unit of D1 contains 1 mg of X and 2 mg of Y. One unit of D2 contains 2 mg of X and 3 mg of Y
Suppose you choose a units of D1 and b units of D2. Then you get a mg of X and 2a mg of Y (from D1) and 2b mg of X and 3b mg of Y from D2. So you get a total of a + 2b mg of X and 2a + 2b mg of Y. What you want is
a + 2b = 13
2a + 3b = 22
Multiply first equation by 2 throughout: 2a + 4b = 26. Subtract the second equation to get b = 4. That gives a + 8 = 13, so that a = 5.
Answer: Take 5 units of D1 and 4 units of D2. Check for yourself that you get 13 mg of X and 22 mg of Y, total, by doing so. (I have!)
Dattaprabhakar (Dr. G.)