Q. 67 - Precalculus Enhanced with Graphing Utilities

Question

A doctor’s prescription calls for a daily intake containing 40 milligrams (mg) of vitamin C and 30 mg of vitamin D. Your pharmacy stocks two liquids that can be used: one contains 20% vitamin C and 30% vitamin D, the other 40% vitamin C and 20% vitamin D. How many milligrams of each compound should be mixed to fill the prescription?

Step-by-Step Solution

Answer

It is necessary to mix 75mg of the first compound and 50 mg of the second compound

Step 1: Given information

Your pharmacy stocks two liquids that can

be used: one contains 20% vitamin C and 30% vitamin D, the other 40% vitamin C and 20% vitamin D.

Step 2: Find the equations

Let x be the first liquid and y be the second liquid

We are given 

Your pharmacy stocks two liquids that can

be used: one contains 20% vitamin C and 30% vitamin D,

Hence 

$$\begin{gathered} 0.2x+0.4y=40 \\ 2x+4y=400 \left(1\right) \end{gathered}$$

The other 40% vitamin C and 20% vitamin D.

Hence we get,

$$\begin{gathered} 0.3x+0.2y=30 \\ 3x+2y=300 \left(2\right) \end{gathered}$$

Step 3: Solve the equation

To solve the equation 

Multiply equation 2 by 2 and subtract it from 1

We get,

$2x+4y=400-6x-4y=-600-4x=-200$

Therefore we get,

$x=50$

Also we find the value of y

$$\begin{gathered} 2x+4y=400 \\ 2\left(50\right)+4y=400 \\ 4y=300 \\ y=75 \end{gathered}$$

Step 4: Conclusion

It is necessary to mix 75mg of the first compound and 50 mg of the second compound