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📘intermediate algebra review

key term - Work-Rate Problems

Citation:

Definition

Work-rate problems are a type of application problem that involve determining the time required to complete a task or the amount of work that can be done in a given time, based on the rate at which the work is being performed. These problems are often encountered in the context of solving rational equations.

5 Must Know Facts For Your Next Test

  1. Work-rate problems often involve the use of the formula: Work = Rate × Time, where work is the total amount of a task completed, rate is the speed or pace at which the task is being performed, and time is the duration over which the task is carried out.
  2. In work-rate problems, the relationship between the variables is often one of inverse variation, meaning that as the rate increases, the time required to complete the task decreases, and vice versa.
  3. Solving work-rate problems typically involves setting up a rational equation that represents the relationship between the variables, and then solving the equation to find the unknown quantity.
  4. Work-rate problems can involve multiple people or machines working together, in which case the overall rate is the sum of the individual rates.
  5. Efficiency and productivity are important considerations in work-rate problems, as they can affect the overall rate at which a task is completed.

Review Questions

  • Explain how the formula Work = Rate × Time is used in the context of work-rate problems.
    • The formula Work = Rate × Time is the foundation for work-rate problems. It represents the relationship between the total amount of work completed, the rate at which the work is being performed, and the time it takes to complete the task. By rearranging this formula, you can solve for any of the three variables if the other two are known. For example, if you know the total work and the rate, you can solve for the time required to complete the task. This formula is essential for setting up and solving the rational equations that are characteristic of work-rate problems.
  • Describe the role of inverse variation in work-rate problems.
    • Work-rate problems often involve an inverse variation relationship between the rate and time variables. This means that as the rate increases, the time required to complete the task decreases, and vice versa. This inverse relationship is reflected in the work-rate formula, where rate and time are in the denominator and numerator, respectively. Understanding this inverse variation is crucial for setting up and solving the rational equations in work-rate problems, as it allows you to determine how changes in one variable will affect the other variable.
  • Analyze how work-rate problems can be extended to situations involving multiple people or machines working together.
    • In work-rate problems with multiple people or machines working together, the overall rate at which the task is completed is the sum of the individual rates. This is because the total work is being divided among the different workers or machines, and the time required to complete the task is the same for all of them. To solve these types of problems, you would need to set up a rational equation that incorporates the individual rates and the total work to be completed, and then solve for the unknown variable, such as the time required or the total work done. Considering efficiency and productivity factors can also be important in these multi-worker or multi-machine scenarios.