Pharmacy A doctor's prescription calls for the creation of...

Pharmacy 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 two powders that can be used to make these pills: One contains $20 \%$ vitamin $\mathrm{B}_{12}$ and $30 \%$ vitamin $\mathrm{E}$, the other $40 \%$ vitamin $\mathrm{B}_{12}$ and $20 \%$ vitamin E. How many units of each powder should be mixed in each pill?

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Key Concepts

Percentage Composition

Percentage composition refers to the ratio of a specific component to the total quantity of a substance, expressed as a percentage. It is essential in these problems for converting between the quantities of substances and their contribution to the desired overall mixture, enabling the formulation of accurate equations for the problem.

Mixture Problems

Mixture problems involve combining different substances with known compositions to obtain a final product that meets specific criteria. This type of problem typically deals with finding the correct quantities of each component whose weighted mixture results in a specified overall concentration or composition.

Systems of Linear Equations

In problems involving mixtures, systems of linear equations are used to represent the constraints imposed by the different components in the mixture. Each equation corresponds to a balance of a particular ingredient, and solving these equations simultaneously provides the necessary proportions of the components.

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