Question

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

Short answer

Mix 50 mg of the first liquid and 75 mg of the second liquid.

Step-by-step solution

- Define Variables

Let the amount of the first liquid be denoted by x milligrams, and the amount of the second liquid be denoted by y milligrams.

- Set Up Equations

Create two equations based on the content of vitamins in each liquid and the total required amount of each vitamin. For vitamin C: 0.20x + 0.40y = 40 For vitamin D: 0.30x + 0.20y = 30

- Simplify the Equations

Simplify the equations if possible. For vitamin C: 2x + 4y = 400 For vitamin D: 3x + 2y = 300

- Solve the System of Equations

Use methods such as substitution or elimination to solve the simplified system of equations. Let's use the elimination method: First, multiply the first equation by 3 and the second equation by 2 to align the coefficients of x: 6x + 12y = 1200 6x + 4y = 600 Next, subtract the second equation from the first: \( (6x + 12y) - (6x + 4y) = 1200 - 600 \) which simplifies to: 8y = 600 y = 75

- Solve for Second Variable

Now substitute y = 75 back into one of the original equations:2x + 4(75) = 400 Simplify: 2x + 300 = 400 2x = 100 x = 50

- Verify the Solution

Verify the solution by checking both conditions: For vitamin C: 0.20(50) + 0.40(75) = 10 + 30 = 40 (Correct) For vitamin D: 0.30(50) + 0.20(75) = 15 + 15 = 30 (Correct)

Key concepts

linear equations

Linear equations play a crucial role in solving mixture problems like the one in this exercise. They represent relationships between different quantities with a constant rate of change. In this problem, linear equations help us equate the amounts of vitamins from different liquid sources.
We create two linear equations to represent the total milligrams of vitamins that need to be mixed. Here’s how we formulate them:

• For vitamin C: The equation 0.20x + 0.40y = 40 means that the required milligrams of vitamin C is a sum of 20% from liquid x and 40% from liquid y.
• For vitamin D: The equation 0.30x + 0.20y = 30 indicates that the required milligrams of vitamin D come from a mix of 30% of x and 20% of y.

These equations let us figure out exactly how much of each liquid we need to meet the vitamin requirements.

substitution method

The substitution method is one way to solve systems of linear equations. This method involves solving one equation for one variable and then substituting that value into the other equation. Let’s illustrate this with our example.
Firstly, we solve one of the equations for one variable. For instance, if we solve the first equation for x, it would look like this:\[ x = \frac{400 - 4y}{2} \]
Next, we substitute this expression for x into the second equation: \[ 3\left( \frac{400 - 4y}{2} \right) + 2y = 300 \]
This substitution turns one equation in two variables into a single equation in one variable, which is easier to solve. The solution for y can be substituted back into the equation for x to find its value. Although the substitution method is practical, it might be more complex than the elimination method for this example.

elimination method

The elimination method is another effective technique used to solve systems of linear equations. It involves aligning the coefficients of one variable, then adding or subtracting the equations to eliminate that variable.
In our example, we adjust the equations to match the coefficients of x:

• Multiply the first equation by 3: 6x + 12y = 1200.
• Multiply the second equation by 2: 6x + 4y = 600.

We then subtract the second equation from the first:\[ 6x + 12y - (6x + 4y) = 1200 - 600 \]
Resulting in:\[ 8y = 600 \]
Solve for y:\[ y = \frac{600}{8} = 75 \]
We substitute y back into one of the original equations to solve for x:\[ 2x + 4(75) = 400 \]
leading to:\[ 2x + 300 = 400 \]
Finally, solve for x:\[ 2x = 100 \]
\[ x = 50 \]
This systematic approach is often preferable, as it provides a straightforward path to the solution.

mixture problems

Mixture problems involve combining different substances to form a mixture that meets specific criteria. In our pharmacy example, we need to derive a solution that gives the patient the correct amounts of vitamin C and vitamin D.
To solve mixture problems efficiently, follow these steps:

• Define your variables: Decide what quantities (x and y) you need to find.
• Create equations: Develop linear equations based on the given information about the substances being mixed.
• Simplify the equations: If possible, simplify the equations to make them easier to solve.
• Choose a solution method: Use either the substitution or elimination method to find the values of x and y.
• Verify your solution: Always substitute the values back into your original equations to ensure they fit all criteria.

Understanding these steps will help you tackle any mixture problem confidently. With practice, you will see how logical and methodical these problems are!