Mixing Nuts A store sells cashews for 5.00 per pound and...

Mixing Nuts $\quad$ A store sells cashews for $\$ 5.00$ per pound and peanuts for $\$ 1.50$ per pound. The manager decides to mix 30 pounds of peanuts with some cashews and sell the mixture for $\$ 3.00$ per pound. How many pounds of cashews should be mixed with the peanuts so that the mixture will produce the same revenue as would selling the nuts separately?

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Key Concepts

Revenue Equivalence

Revenue equivalence is the principle of ensuring that the total revenue obtained from selling a mixture at a set price is equal to the combined revenue that would be earned by selling its individual components at their original prices. This concept is crucial when adjusting mixtures to maintain a desired pricing outcome.

Algebraic Modeling

Algebraic modeling involves translating a real-world situation into mathematical expressions or equations. In mixture problems, this often means setting up equations that represent the total cost, weight, and revenue contributions of each component, which can then be solved for the unknown quantities.

Mixture Problems

Mixture problems involve combining different components with distinct properties to form a new mixture that meets a specific target characteristic, such as an average cost or concentration. These problems require determining the proportion of each component needed to achieve the desired outcome.

Weighted Average

The weighted average is a calculation that factors in the relative contribution of each component in a mixture, where each component is multiplied by its respective weight before summing and then dividing by the total weight. This concept is essential in determining the overall value, such as the average price per unit, of a mixture.

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