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# Basic work/rate problems and algebra

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Basic work/rate problems and algebra [#permalink]  31 Jan 2013, 10:29
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Hi,

Some of the basic work/rate problems keep bugging me. I must have missed a couple of fundamental algebra shortcuts.

e.g if I want this to be expressed in terms of t: $$\frac{1}{t} = \frac{1}{x} + \frac{1}{y}$$
The answer is supposed to be $$t = \frac{yx}{y-z}$$

Could someone please walk through the process and rules here.

Cheers
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Re: Basic work/rate problems and algebra [#permalink]  31 Jan 2013, 20:14
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Expert's post
Monkeybiz wrote:
Hi,

Some of the basic work/rate problems keep bugging me. I must have missed a couple of fundamental algebra shortcuts.

e.g if I want this to be expressed in terms of t: $$\frac{1}{t} = \frac{1}{x} + \frac{1}{y}$$
The answer is supposed to be $$t = \frac{yx}{y-z}$$

Could someone please walk through the process and rules here.

Cheers

Now that you know how to algebraically modify the equation, let me talk about what you can do in work-rate problems in particular.

Instead of learning up the equation and how to modify it algebraically, you can solve work-rate questions using some basic fundamentals.

Work Rate problems are based on two and only two concepts:
2. Work done = Rate * Time (much like Distance = Speed * Time).
W = R*T

Let me first discuss how you get 1/T = 1/X + 1/Y (T - time taken while working together, X - time taken by X alone, Y - time taken by Y alone).

Say work W is 'paint 1 wall'
1 = R*T
So Rate = 1/Time
Since rates are additive, Rate of working together = Rate of X alone + Rate of Y alone
1/T = 1/X + 1/Y
Now, you plug in the values of X and Y and get the value of 1/T which you can flip to get the value of T. Say, if
X = 2 hrs and Y = 4 hrs, then 1/T = 1/2 + 1/4 = 3/4
So T = 4/3 hrs

Actually, if you keep the two points given above in mind, you should face no problems in work-rate questions.

If I paint half a wall in an hour and if you paint half a wall in an hour (recall, rate of work is just the speed of doing work), if we both work together on a wall, we will finish the wall in an hour.
If my rate of work is 1/2 wall/hour and yours is 1/2 wall/hour, our total rate of work is 1/2 + 1/2 = 1 wall/hour (that is what we mean by 'rates are additive')

Now, if the question is: How many walls will we paint together in 8 hours?

Work = Rate*Time
Work = 1*8 = 8 walls

Use these two concepts and you will be able to solve most questions.

The basic questions of work rate are of the following form:
If A, working independently, completes a job in 10 hours and B, working independently, completes a job in 5 hours, how long will they take to complete the same job if they are working together?

Since A completes a job in 10 hours, his rate of work is 1/10th of the job per hour. B's rate of work is 1/5th of the job per hour.
Their combined rate of work would then be 1/10 + 1/5 = 3/10th of the job per hour.
As we said before, Work done = Rate * Time
1 = 3/10 * T (because 1 job has to be done)
or T = 10/3 hours.
This implies that A and B will together take 3.33 hours to do the job.
Note: Time taken when A and B work together will obviously be less than time taken by A or B when they are working independently.
Note: Time is not additive. In the question above, you obviously cannot say that A and B will together take 10+5 = 15 hrs to complete the job. Working together, they will take less time than either one of them working alone takes.

Try using these two concepts to solve your work-rate problems... Post specific questions if issues arise...

Also check out this post on work rate problems: http://www.veritasprep.com/blog/2011/03 ... -problems/
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Re: Basic work/rate problems and algebra [#permalink]  31 Jan 2013, 16:28
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1/t=1/x+1/y
Implies 1/t = (y+x)/xy. (Add using lowest common multiple - is this the area of concern?)
Implies t=xy/(x+y). (Cross multiply)

Does that help?
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Re: Basic work/rate problems and algebra [#permalink]  02 Feb 2013, 12:41
jumsumtak: Yeah, finding LCM with letters rather than numbers confused me. Thanks for your clarification.

VeritasPrepKarishma: Great thanks. Must have missed your blog, what an amazing resource.
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Re: Basic work/rate problems and algebra [#permalink]  15 Sep 2014, 12:22
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Re: Basic work/rate problems and algebra   [#permalink] 15 Sep 2014, 12:22
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