Question

a doctors prescription calls for a daily intake containing 60 mg of vitamin c and 40 mg of vitamin d. your pharmacy stocks two compounds that can be used: one contains 40% vitamin c and 40% vitamin d, the other 40% vitamin c and 20% vitamin d. how many milligrams of each compound should be mixed to fill the prescription? to fill the prescription, mix mg of the first compound with mg of the second compound.

Explanation:

Step1: Set up equations

Let $x$ be the amount (in mg) of the first compound and $y$ be the amount (in mg) of the second compound.
For vitamin C: $0.4x + 0.4y=60$.
For vitamin D: $0.4x+0.2y = 40$.

Step2: Simplify the vitamin - C equation

Divide the equation $0.4x + 0.4y=60$ by $0.4$ to get $x + y=150$, so $x=150 - y$.

Step3: Substitute into the vitamin - D equation

Substitute $x = 150 - y$ into $0.4x+0.2y = 40$.
$0.4(150 - y)+0.2y=40$.
Expand: $60-0.4y + 0.2y=40$.
Combine like - terms: $60-(0.4y - 0.2y)=40$, $60 - 0.2y=40$.
Subtract 60 from both sides: $-0.2y=40 - 60=-20$.

Step4: Solve for $y$

Divide both sides by $- 0.2$: $y=\frac{-20}{-0.2}=100$.

Step5: Solve for $x$

Substitute $y = 100$ into $x=150 - y$, so $x=150 - 100 = 50$.

Answer:

50, 100