A doctor's prescription calls for the creation of pills that...

A doctor's prescription calls for the creation of pills that contain 12 units of vitamin $\mathrm{B}_{12}$ and 12 units of vitamin E. Your pharmacy stocks three powders that can be used to make these pills: one contains $20 \%$ vitamin $B_{12}$ and $30 \%$ vitamin E; asecond. $40 \%$ vitamin $\mathrm{B}_{12}$ and $20 \%$ vitamin $\mathrm{E}$ and a third, $30 \%$ vitamin $B_{12}$ and $40 \%$ vitamin E. Create a table showing the possible combinations of these powders that could be mixed in each pill. Hint: 10 units of the first powder contains $10(0.2)=2$ units of vitamin $\mathrm{B}_{12}$

Key Concepts

Feasible Combinations and Constraints

When combining ingredients to meet specific targets, not every mathematical solution may be practical. Understanding the feasible combinations means considering the constraints—such as the required minimal or exact amounts of components—and then identifying all mixtures that meet the prescribed requirements.

Mixture Problems

This concept focuses on combining different substances in specific ratios or amounts to achieve a desired overall composition. In these problems, each ingredient contributes certain proportions of components (such as vitamins, in this case), and the challenge is to determine the amounts that need to be mixed to meet a set of target values.

Linear System of Equations

In mixture problems, the relationships between the amounts of different substances and the resulting nutritional or chemical contents are often expressed as a set of linear equations. Each equation represents a balance of components (like vitamins) and solving the system allows us to find all possible combinations that satisfy the constraints.

Concentration and Percentage Calculations

This concept involves using the given percentages or concentrations of a substance within an ingredient to calculate the actual amount of that substance present in a specified quantity of the ingredient. It is essential for converting between percentage data and the actual contribution to the overall mixture.

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