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

Mixing Nuts 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 selling the nuts separately?

Key Concepts

Weighted Average

Weighted average is a method of calculating an overall average by assigning different weights to data points based on their significance or proportion. In mixture problems, it is used to determine the resulting average price or concentration by proportionally considering the cost or concentration of individual components.

Mixture Problems

Mixture problems involve combining substances or items with different characteristics (such as cost, concentration, or percentage composition) to form a new mixture that meets certain criteria. They often require setting up equations that reflect the balance of quantities and values, allowing for the determination of unknown amounts.

Linear Equations

Linear equations are algebraic expressions where variables are raised only to the first power. They are used in mixture problems to model relationships between the quantities and prices of different components, ultimately providing a straightforward method to solve for an unknown quantity by setting up an equation based on the conservation of cost or mass.

Revenue Equivalence

Revenue equivalence refers to the concept of ensuring that the overall revenue generated by a combined mixture equals the total revenue that would have been obtained if the components had been sold separately. This approach is crucial in problems where the goal is to match or optimize financial outcomes through strategic blending of different items.

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