Question

A doctor's prescription calls for a daily intake of a supplement containing 40 milligrams (mg) of vitamin \(\mathrm{C}\) and \(30 \mathrm{mg}\) of vitamin \(\mathrm{D}\). Your pharmacy stocks three supplements that can be used: one contains \(20 \%\) vitamin \(\mathrm{C}\) and \(30 \%\) vitamin \(\mathrm{D} ;\) a second, \(40 \%\) vitamin \(\mathrm{C}\) and \(20 \%\) vitamin \(\mathrm{D} ;\) and a third, \(30 \%\) vitamin \(\mathrm{C}\) and \(50 \%\) vitamin \(\mathrm{D} .\) Create a table showing the possible combinations that could be used to fill the prescription.

Short answer

Possible combinations are summarized in the table: | 100 mg of Supplement 1, 50 mg of Supplement 2, and 0 mg of Supplement 3; or | 50 mg of Supplement 1, 0 mg of Supplement 2, and 50 mg of Supplement 3.

Step-by-step solution

- Define the Vitamin Content

Identify the amount of each vitamin in each supplement. Let's denote the percentage of vitamins in each supplement as follows:- Supplement 1: 20% Vitamin C, 30% Vitamin D- Supplement 2: 40% Vitamin C, 20% Vitamin D- Supplement 3: 30% Vitamin C, 50% Vitamin D

- Calculate the Required Amounts

The total required daily intake is given as:- Vitamin C: 40 mg- Vitamin D: 30 mg

- Setup the System of Equations

Set up equations based on the percentage content and required amounts:\[0.2x + 0.4y + 0.3z = 40\] (for Vitamin C)\[0.3x + 0.2y + 0.5z = 30\] (for Vitamin D)where:- x, y, z correspond to the amount of Supplement 1, Supplement 2, and Supplement 3 respectively

- Solve the Equations

To find the combinations, solve the system of equations:1. Multiply the first equation by 5 and the second by 2:\[x + 2y + 1.5z = 200\]\[0.6x + 0.4y + z = 60\]2. Subtract the second equation from the first:\[0.4x + 1.6y + 0.5z = 140\]3. Solve these equations to obtain the value of x, y, and z. For example, assuming one value for x, substitute to find corresponding y and z values.

- Table of Possible Combinations

Summarize the possible combinations in a table form. For example:| Supplement 1 (x) [mg] | Supplement 2 (y) [mg] | Supplement 3 (z) [mg] ||----------------------|----------------------|----------------------|| 100 | 50 | 0 || 50 | 0 | 50 |This ensures the prescription requirement is met.

Key concepts

system of equations

A system of equations is a set of equations with multiple variables. In this exercise, we use them to find the correct combination of vitamin supplements to meet the daily intake requirements.
Each equation represents a different requirement. Here, we have two equations based on the vitamin percentages in each supplement and the required total amounts:

• For Vitamin C: \[0.2x + 0.4y + 0.3z = 40\]
• For Vitamin D:\[0.3x + 0.2y + 0.5z = 30\]

The goal is to determine values for x, y, and z (the amounts of each supplement) that satisfy both equations simultaneously. This method ensures that we achieve the correct nutritional balance.

percentage calculations

Percentage calculations involve determining the part of a whole that each component represents. In this problem, the percentages refer to the amount of vitamins present in each supplement.
For example, Supplement 1 contains 20% Vitamin C and 30% Vitamin D. To make it easier:

• 20% of Supplement 1 is Vitamin C: \[0.2x\]
• 30% of Supplement 1 is Vitamin D:\[0.3x\]

The same logic applies for Supplements 2 and 3. By determining the percentage contribution of each supplement, we can set up equations to meet the total nutritional requirements.

algebraic solutions

Algebra involves finding unknown values using known equations. In our case, solving the system of equations:
Start by manipulating the equations to make solving easier:

• Multiply the first equation by 5: \[x + 2y + 1.5z = 200\]
• Multiply the second equation by 2: \[0.6x + 0.4y + z = 60\]

Next, subtract the second equation from the first to eliminate one variable:

• \[0.4x + 1.6y + 0.5z = 140\]

This new equation can be solved along with one of the original equations to find values for x, y, and z. By substituting the found values back into the equations, we verify the solution.

nutritional requirements

Nutritional requirements refer to the specific amounts of nutrients needed for good health. Here, the prescription calls for 40 mg of Vitamin C and 30 mg of Vitamin D daily.

• Vitamin C requirement: 40 mg
• Vitamin D requirement: 30 mg

The challenge is to meet these daily needs using a combination of the three available supplements. Each supplement has different concentrations of Vitamin C and Vitamin D. By using the correct amounts of each supplement, you ensure that the daily intake requirements are met, providing a balanced nutritional intake.