Write an equation for each and solve. See Example 3 . It...

Write an equation for each and solve. See Example $3 .$ It would take Mei twice as long as Ting to make decorations for a party. If they worked together, they could make the decorations in 40 min. How long would it take Mei to make the decorations by herself?

Key Concepts

Work Rate

Work rate is the measure of how much of a task can be completed in a unit of time. In work-related problems, each individual is assigned a rate, typically expressed as a fraction of the task completed per unit of time. Understanding work rate is crucial because it allows for the comparison of different workers' efficiencies and facilitates the setup of equations based on the work done over time.

Combined Work Rate

The combined work rate concept refers to the scenario where multiple individuals work together to complete a task. The key principle is that the total work rate is the sum of the individual work rates. This additive property allows one to calculate the overall time needed to finish a task when working collaboratively by setting up an equation where the sum of the individual rates equals the reciprocal of the total time taken when working together.

Relative Time and Efficiency

Relative time and efficiency involve comparing the rates at which different individuals perform a task. In problems where one person is said to take a multiple of the time compared to another, the relationship translates into their work rates being reciprocally related. For instance, if one person takes twice as long as another to complete a task, the faster individual’s work rate is double that of the slower individual. This concept is essential to correctly formulate relationships between different workers’ speeds.

Setting Up Algebraic Equations

Algebraic equations are used to model work rate problems by translating the problem's narrative into mathematical relationships. Variables are introduced to represent unknown times or rates, and equations are constructed based on the principle that work equals rate multiplied by time. Solving these equations involves algebraic manipulation and is key to determining the individual times required to complete the task. This method provides a systematic approach to solving a variety of work-related problems.

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