Write an equation for each and solve. A faucet can fill a...

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

Key Concepts

Work Rate Problems

Work rate problems involve situations where tasks are completed by one or more agents working over time. The key idea is to express each agent's contribution as a rate (for example, the fraction of the job completed per unit time) and use these rates to determine how long it takes to complete the entire task when they work together or against each other.

Net Rate Calculation

The net rate calculation is used when multiple processes affect the task simultaneously. In cases where some processes work towards completing the task and others work against it, their rates are added algebraically (with subtracting the negative rates) to determine the overall rate of progress. This net rate then helps in finding the total time required to complete the task.

Equation Setup for Work Problems

Setting up an equation in work problems typically involves multiplying the rate by the time to equate to the full task (often taken as one whole job). This step involves converting given times into rates, combining them appropriately (addition or subtraction depending on the problem), and solving for the unknown time that satisfies the equation.

Transcript

00:01 A faucet can fill a tub in 12 minutes and the drain can completely get rid of the water or drain the tub in 30 minutes.

00:09 And we're asked, so if the faucet is on and the drain leaks the water out, how long would it take to completely fill the tub? well, let's look at the rates of these individual items, the faucet and the drain.

00:26 So the rates in this case would be the amount of job done per hour.

00:33 So let's look at the faucet, we get one job per 12 minutes.

00:41 Or actually, we should probably use minutes, keep it consistent in this case, and set it equal to a denominator of one minute.

00:52 So the numerator would be x, and now we can cross multiply to solve for x.

00:57 So we get 12x equals 1 or x equals 112th.

01:03 So the rate of the faucet is 1 twelfths job per, and the drains rate is one job per 30 minutes.

01:16 And we're trying to find the same thing.

01:18 We're trying to find how much of the job it completes in one minutes.

01:22 So y divided by 1 minute.

01:24 Cross multiply once again.

01:27 And we get 30 y equals 1.

01:31 Or y equals one dirtieth job per minute.

01:38 This is the drains.

01:39 This is the drains rate.

01:44 So now to find if both of them were on, normally we would, if both of them were accomplishing the same task, we would add them together.

01:52 But since the drain is decreasing from the task, decreasing the water, which is what the faucet is doing, we subtract them from each other.

02:02 So we take the difference, which is 112 minus 1 .30th because water is moving in opposite, they're accomplishing opposite tasks.

02:12 The drain is making it harder for the faucet to fill the tub to accomplish its job...

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