📘intermediate algebra review

key term - Weighted Average

Definition

A weighted average is a type of average where different values in a dataset are assigned different weights or levels of importance. It is commonly used in mixture applications to calculate the overall composition or characteristic of a mixture based on the individual components and their respective proportions.

5 Must Know Facts For Your Next Test

  1. The weighted average formula is: $\sum_{i=1}^{n} w_i x_i / \sum_{i=1}^{n} w_i$, where $w_i$ is the weight or proportion of the $i$-th component and $x_i$ is the value of the $i$-th component.
  2. Weighted averages are useful in mixture applications because they allow you to account for the different proportions or concentrations of the individual components.
  3. Solving mixture applications with systems of equations often involves setting up a weighted average equation to represent the overall composition or characteristic of the mixture.
  4. The weights used in a weighted average calculation are typically based on the relative amounts or proportions of the components in the mixture.
  5. Weighted averages can be used to calculate various mixture properties, such as density, concentration, pH, or any other characteristic that depends on the individual component values and their respective proportions.

Review Questions

  • Explain how a weighted average is used to solve mixture applications with systems of equations.
    • In mixture applications, a weighted average is used to represent the overall composition or characteristic of the mixture based on the individual component values and their respective proportions. By setting up a weighted average equation as part of a system of equations, you can solve for the unknown variables, such as the amounts or concentrations of the mixture components. The weights used in the weighted average calculation are typically the relative proportions of the components in the mixture.
  • Describe the key differences between a weighted average and an arithmetic mean, and explain why a weighted average is more appropriate for mixture applications.
    • The key difference between a weighted average and an arithmetic mean is that a weighted average assigns different levels of importance or weights to the individual values, whereas an arithmetic mean treats all values equally. In mixture applications, a weighted average is more appropriate because the individual components have different proportions or concentrations in the mixture, and these differences need to be accounted for to accurately represent the overall composition or characteristic of the mixture. The weighted average formula allows you to incorporate these varying proportions, whereas the arithmetic mean would not accurately reflect the true nature of the mixture.
  • Analyze how the use of a weighted average in a system of equations can help you solve for the unknown variables in a mixture application, and explain the significance of the weights used in the calculation.
    • When solving mixture applications with systems of equations, the use of a weighted average equation is crucial because it allows you to represent the overall composition or characteristic of the mixture based on the individual component values and their respective proportions. By incorporating the weights, which correspond to the relative amounts or concentrations of the components, the weighted average equation provides a more accurate and representative model of the mixture. This, in turn, enables you to solve for the unknown variables, such as the quantities or concentrations of the individual components, by solving the system of equations that includes the weighted average equation. The significance of the weights lies in their ability to capture the relative importance or influence of each component on the overall mixture, which is essential for obtaining meaningful and accurate solutions in mixture applications.