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

Write an equation for each and solve. See Example $3 .$ A faucet can fill a tub in 12 minutes. The leaky drain can empty the tub in 30 minutes. If the faucet is on and the drain is leaking, how long would it take to fill the tub?

Key Concepts

Work Rate

Work rate problems involve determining how long it takes a process to complete a task when given the rate at which work is done. In these problems, a rate is typically expressed as the fraction of the task completed per unit time. This concept is crucial for assessing how individual contributions or detriments combine over time to impact overall progress.

Net Rate

The net rate concept involves combining multiple rates that work simultaneously, often with one process adding to progress (such as filling) and another subtracting from it (such as emptying). By adding the positive and negative rates together, one can determine the overall speed at which the task is being completed.

Fractional Rates and Unit Conversion

When dealing with rates, it is common to express the work completed as a fraction of the total task per unit time. This may require converting between different units or simplifying fractional expressions to facilitate the calculation of a net rate or the total time required to complete a task.

Equation Setup and Solving

Setting up an equation in work rate problems involves representing the contribution of each process (filling and emptying) and combining them to equal the net progress. Solving the equation typically requires isolating the variable representing time, which is often done by calculating the reciprocal of the net rate once the rates have been appropriately combined.

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