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Work Problems

Work problems involve calculating how long it takes for individuals or groups to complete a task, either separately or together. The standard model is:

$$ \text{Work} = \text{Rate} \times \text{Time} $$

Basic Work Rate:

If a person or machine can complete a task in $x$ hours, the rate of work is:

$$ r = \frac{1}{x} $$

Combined Work Rate:

When multiple agents work together:

$$ r_{\text{total}} = r_1 + r_2 + \dots $$

Man-Hour or Man-Day Problems:

Another type of work problem involves man-hours or man-days as a unit of work. These problems assume the total work remains the same, and the number of workers and time vary.

The equation is based on the principle that:

$$ \frac{\text{Man} \times \text{Time}}{\text{Work}}_1 = \frac{\text{Man} \times \text{Time}}{\text{Work}}_2 $$

This can be applied as:

This method is especially useful when the number of workers or amount of work changes across different scenarios.

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