Painting a House Three painters (Beth, Bill, and Edie), working together, can paint the exterior

Painting a House Three painters (Beth, Bill, and Edie),...

Key Concepts

Work Rate

This concept refers to the amount of work completed per unit of time. In problems like these, each agent has an individual work rate, often expressed as the fraction of the total job completed in one hour. Understanding work rate is critical because it allows the conversion of time requirements into productive capacity that can be added or compared.

Combined Work Rates

When multiple agents work together, their individual work rates add up to determine the overall work rate for the team. This principle is essential in solving problems where several people or machines collaborate to complete a task, as it enables the calculation of the total work done in a specific timeframe.

Systems of Equations

Many work problems require setting up and solving systems of equations to find unknown individual work rates. Each condition provided in the problem translates into an equation that represents a combination of individual rates over a given period. Solving these simultaneous equations yields the rate for each agent.

Reciprocal Relationship Between Time and Work Rate

This key concept outlines that the time taken to complete a job is the reciprocal of the work rate, assuming a constant rate. In other words, if an agent can complete a certain portion of a job per hour, the total time to finish the job when working alone is found by taking the inverse of that fraction.

Sequential Work and Partial Completion

In many work problems, the work is done in stages, either because certain workers leave or join at different times. This concept involves accounting for portions of the work completed in different time intervals and combining these to figure out the remaining workload, which is critical for setting up accurate equations.

Transcript

00:01 So for this problem, we're given information about beth, bill, and edie and their house painting capabilities.

00:08 And we're asked to identify how long it would take each one of them to paint a house on their own, based on the ways they can paint houses interacting with each other.

00:23 So i'm going to assign a variable to each of these people.

00:27 Beth will be x.

00:28 Bill will be y and ed will be z.

00:33 I would use letters of their name, except that e is its own number when it's a variable.

00:39 And beth and bill both start with b.

00:41 So i'll use x, y, and z just to keep things clear, as that's pretty typical for systems of equations.

00:49 The other thing is what these variables represent.

00:52 So that variable is going to represent how many houses per per hour that individual can paint.

01:04 So we're looking at houses per hour.

01:08 And that's just their rate of painting houses.

01:12 And so we know that when we have all of them together and they're working for 10 hours, they complete one house.

01:22 So that's when beth is working for 10 hours, and bill is working for 10 hours.

01:29 And edie is working for 10 hours.

01:30 And edie is working for 10 hours.

01:32 We know all of this gives us one house complete.

01:39 We also know that without bess, with just bill and edi, it takes 15 hours.

01:47 So 15 hours of bill combined with 15 hours of edi also gives us one complete house.

01:59 And the last thing that we know is that if, if all three of them work for four hours.

02:10 So that's four of beth, four of bill, and four of ead.

02:21 And then we know that ead steps out and beth and bill work for an additional eight hours.

02:33 So that's eight more for beth and eight more for bill.

02:37 That that gives us one complete house.

02:41 So i'm going to rewrite these equations over here just so that i can combine some of this down here into one.

02:52 That way we have all of our equations really set and in the same form.

03:04 So the first two equations can stay the same.

03:07 They're already in that similar format.

03:11 For the second one, i could write.

03:13 Write a 0x at the beginning to indicate that beth isn't working, but that's just kind of implied by the fact that it's not there.

03:23 So here we have a total of 12x, a total of 12y, but only 4z.

03:41 And looking at the system of equations, there's a number of things you could do.

03:46 You could pull some substitution using this second equation, knowing that it would be pretty easy to solve for y or z in terms of the other.

03:59 But i'm actually going to use elimination with the first two equation, or the first equation and the last equation.

04:07 So i can see that to eliminate, i could get to a 60 by multiplying the first equation by six on both sides, and by multiplying my last equation by five on both sides.

04:24 So what that gives me is 60x plus 60y plus 60 z equals 6.

04:34 And my second equation gives me 60x plus 60y plus 20 z is equal to 5.

04:47 And then i'm going to subtract these two equations.

04:50 So that gives me 0x, 0y, 40z equals 1...

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