Problem 14 - Algebra and Trigonometry with Analytic Geometry

Question

A pharmacist is to prepare 15 milliliters of special eye drops for a glaucoma patient. The eye-drop solution must have a \(2 \%\) active ingredient, but the pharmacist only has \(10 \%\) solution and \(1 \%\) solution in stock. How much of each type of solution should be used to fill the prescription?

Step-by-Step Solution

Answer

Use 1.67 ml of the 10% solution and 13.33 ml of the 1% solution.

Step 1: Define Variables

Let \( x \) be the amount of the \(10\%\) solution and \( y \) be the amount of the \(1\%\) solution needed to prepare the \(15\) milliliters of \(2\%\) eye drops.

Step 2: Set Up System of Equations

The total volume of the solution should be \(15\) milliliters: \ \[ x + y = 15 \] The equation for the active ingredient concentration becomes: \[ 0.1x + 0.01y = 0.02 imes 15 \] Simplifying, this gives us: \ \[ 0.1x + 0.01y = 0.3 \]

Step 3: Solve the System of Equations

Solve the first equation for \( y \): \ \[ y = 15 - x \] Substitute \( y \) in the second equation: \ \[ 0.1x + 0.01(15 - x) = 0.3 \] Simplify and combine terms: \ \[ 0.1x + 0.15 - 0.01x = 0.3 \] This reduces to: \ \[ 0.09x = 0.15 \] Solve for \( x \): \ \[ x = \frac{0.15}{0.09} = \frac{5}{3} = 1.67 \] Substitute \( x = 1.67 \) back to find \( y \): \ \[ y = 15 - 1.67 = 13.33 \]

Step 4: Check the Solution

Verify the solution by substituting \( x = 1.67 \) and \( y = 13.33 \) into the original concentration equation: \ \[ 0.1(1.67) + 0.01(13.33) \approx 0.167 + 0.133 = 0.3 \] Since both equations are satisfied, the solution is correct.

Key Concepts

system of equations

When dealing with mixture problems in math, such as finding the correct amount of two different solutions to achieve a desired concentration, we often use a system of equations. This method is an excellent tool that helps us solve scenarios involving multiple variables and conditions at once.
For the problem at hand, we have two unknowns: the amount of the 10% solution \(x\) and the amount of the 1% solution \(y\), both needed to create a total of 15 milliliters of a 2% solution.
Here's how it works:

• The first equation comes from the total volume requirement: \[x + y = 15\]. This equation ensures that the combined solution will be exactly 15 milliliters.
• The second equation deals with the active ingredient: \[0.1x + 0.01y = 0.3\]. Here, 0.3 milliliters represent 2% of the 15 milliliters total solution, which is our target concentration.

By solving this system of equations, we can find precise values for \(x\) and \(y\), ensuring both volume and concentration requirements are met. Breaking down complex problems like this with a system of equations simplifies the work and offers a clear path to a solution.

concentration calculation

Calculating the concentration of a solution, especially in pharmacy, is about ensuring that the correct amount of active ingredient is present in a given volume of liquid. This concentration is often expressed as a percentage.
For example, in this problem, we want a 2% active ingredient concentration in a 15 milliliter solution of eye drops. The concentration calculation ensures that the formula: \( \text{active ingredient volume} = \text{total volume} \times \text{desired concentration}\) is maintained.
Specifically, we calculate:

• Active ingredient needed: \(0.02 \times 15 = 0.3\) milliliters.

So, to achieve the desired concentration, this 0.3 milliliters must be a combination of active ingredients from both the 10% and 1% solutions. Proper concentration calculations ensure the medication will be effective and safe, by not exceeding or falling short of the needed active ingredient amount.

pharmacy math

Pharmacy math is crucial in pharmacy practice because it allows pharmacists to accurately compound and dispense medications according to prescriptions.
For pharmacists, it is essential to understand concepts like dilution, concentration, and volume measurements. This problem is a typical example, where precise measurements and calculations ensure that the patient receives the correct dosage of medication.
Important aspects of pharmacy math include:

• Understanding percentages to calculate drug concentrations.
• Using systems of equations to manage multiple ingredients in a solution.
• Applying logical reasoning to ensure patient safety through precise dosages.

Mastering these skills enables pharmacists to adhere to regulations and deliver accurate treatments that effectively address patients' medical needs. By practicing these calculations, aspiring pharmacists can feel confident in their ability to prepare and dispense medications correctly.