One technician can wire a security alarm in 4 h, whereas it takes 6 h for a second technician to

One technician can wire a security alarm in 4 h, whereas it...

One technician can wire a security alarm in $4 \mathrm{h}$, whereas it takes $6 \mathrm{h}$ for a second technician to do the same job. After working alone for $2 \mathrm{h}$, the first technician quits. How long will it take the second technician to complete the wiring?

Key Concepts

Work Rate

Work rate describes the fraction of a task a worker or machine can complete in a unit of time. It is calculated by taking the reciprocal of the total time required to finish an entire job, which allows comparison of efficiency between different agents.

Work, Rate, and Time Relationship

This fundamental concept states that the amount of work completed is equal to the product of the work rate and the time spent working. Understanding this relationship is crucial for determining any one of the three variables when the other two are known.

Fraction of Work Completed

In many work-rate problems, it is essential to calculate the portion of the job that has been finished by a worker over a given time. This involves multiplying the worker’s rate by the time worked to determine the fraction of the total task that is complete.

Remaining Work Calculation

Once a part of the work is completed, the remaining work is found by subtracting the completed fraction from the whole (often represented as 1). This remainder, when combined with the work rate of the available worker, can be used to determine how much additional time is needed to finish the task.

Transcript

00:01 Our word problem states that one technician can wire a security alarm in four hours, and it takes the second technician six hours to do the exact same job.

00:11 After working alone for two hours, the first technician decides to quit.

00:16 How long will it take that second technician to complete the wiring that the first technician started? so we have technician one, so tech number one, and it takes him four hours, and then we have the technician number two, which would take six hours for completion.

00:38 And then we're given that after working alone, tech number two, or tech number one quits after two hours.

00:50 So then we're asked to find how long would it take tech number two to complete the job of the first technician.

00:59 So we're going to start with tech number one.

01:01 And look at part of the job in one unit of time, and in this scenario, it's going to be hours, equals one over the time to complete.

01:20 So in this scenario, we have one over the four hours in which it takes him to complete a job.

01:29 So then we would look at part of the job that he completed in t hours, but we have t hours, which is two.

01:38 So in two hours, how much did he get done already of the job? and that would be part of the job in one hour, which we just solved for, times t, which is going to be two.

01:52 So it would be one over four times two.

01:55 So that would end up being two over four or one over two.

02:04 So now that we have that, we need to look at the second technician...

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