At The Crispy Critter's Head Shop and Patchouli Emporium...

At The Crispy Critter's Head Shop and Patchouli Emporium along with their dried up weeds, sunflower seeds and astrologieal posteards they sell an herbal tea blend. By weight, Type I herbal tea is $30 \%$ peppermint, $40 \%$ rose hips and $30 \%$ chamomile, Type II has percents $40 \%$, $20 \%$ and $40 \%$, respectively, and Type III has percents $35 \%, 30 \%$ and $35 \%$, respectively. How much of each Type of tea is needed to make 2 pounds of a new blend of tea that is equal parts peppermint, rose hips and chamomile?

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Key Concepts

Mixture Problems

Mixture problems deal with combining different substances or solutions with varying properties to achieve a mixture with a desired composition. They require setting up relationships based on the proportions and amounts of each component in the mixtures.

Percent Composition

Percent composition is used to describe the proportion of each ingredient within a mix by weight. This concept allows us to convert percentage values into actual quantities and is essential for determining how much of each component is present in a given amount of mixture.

System of Linear Equations

When solving mixture problems, a system of linear equations is typically established to represent the relationships between the different components. Each equation corresponds to a conservation law for one of the substances (like total weight or a specific ingredient), and solving these equations simultaneously yields the required amounts for each mixture.

Weighted Averages

Weighted averages are used in these problems to calculate the overall composition of the mixture based on the contributions from different components. This concept helps reconcile the differences in concentration levels by adjusting the proportions of each type according to their weight or volume.

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