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Old 06-23-2014, 11:16 AM   #21
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I had a hard time finding this on-line. But I do have the hard copy. Apparently, they are embarrassed by the mistake and buried the link.

http://parade.condenast.com/308009/m...savant/308009/

One of the posters there did a far more concise version of my 'dis-proof' of her numbers:


Quote:
rbttucker

Using Marilyn’s answer, Angelina’s rate is project/10 and Brad’s rate is project/14. In 6 hours Brad would complete 6/14ths of a project Angelina would complete 6/10ths of a project. Together that adds up to 1.02857 of a project.

What am I missing??
And 1.02857 is the same (rounded) as my 648/630 example (LibreOffice says: 1.02857142857143000000).

-ERD50
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Old 06-23-2014, 11:21 AM   #22
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I don't understand. I thought they got it done in 6 hours?
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Old 06-23-2014, 11:21 AM   #23
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I've found over the years that I cannot get excited about a puzzle (especially one that has a clear precise answer) unless I can make lots of money solving it. One of those dirty capitalists I suppose.
In contrast, I like puzzles that have a precise answer. And that usually means math problems.

Though I like engineering, it always involves economic factors (else it would be called science). These bring fuzziness and a lot of debates into the solutions, but it is a fact of life that I have to accept.

Worse is when it involves politics, where the best technical solution is not adopted because it might upset someone in upper management, etc... It was one of the factors I ER'ed, although I have always liked my work and got paid quite decently.
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Old 06-23-2014, 11:35 AM   #24
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I don't understand. I thought they got it done in 6 hours?
Right, combined they get it done in 6 hours, so working together, they could get two projects done in 12 hours.

Where Marilyn gets it wrong is - in her example, Brad is working alone for 4 hours. Well, they can't get the two projects completed at that same rate when only one of them is working for part of the time (the 4 hours Brad works alone). It obviously takes longer when one of them stops working.

So the correct answers are slightly longer than 10 and 14 hours.

She assumed 24 'person-hours' to complete two projects, but that is only if they are both working the whole time. She should have stared working backwards with the 4 hour time delta, and filled in the blanks (as the other solution here have done), to find out how much time each required, rather than making the wrong assumption about '(24-4)/2'

Make sense?


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Old 06-23-2014, 11:49 AM   #25
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Actually no, it does not make sense. I was working from the problem as originally presented in post #1 by NWB:
Without further ado, here's the puzzle.

Two persons, A and B, work together on a project and finish it in 6 hours. If working alone, A finishes it in 4 hours less than the time that B takes. How long does it take A and B to do it individually?
Oh, now I see how I made my mistake! I didn't read the question properly.
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Old 06-23-2014, 12:00 PM   #26
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There is one task to be done so let A be the rate at which A accomplishes the task and let B be the rate at which B accomplishes the task.

Working together, the one task is done with their combined rate in 6 hours:

1) 1/(A+B) = 6

Working separately,

2) 1/A = t (unknown time)
3) 1/B = t + 4 (also unknown, but 4 hours greater than the time A takes)

Doing some algebra with equation 1, A+B = 1/6

Also doing algebra with equations 2 and 3,

A = 1/t
B= 1/(t+4)

This, by the way is just the definition of the rate at which they do the task.

Substituting,

1/t + 1/(t+4) = 1/6

Multiply both sides by 6(t)(t+4) gives:

6t+24 + 6t = t(t+4)

12t + 24 = t^2 +4t

Rearranging,

t^2 - 8t -24 = 0

Using the quadratic equation to solve for t gives two solutions,

t = 4 + 2 sqrt(10)

or

t = 4 - 2 sqrt(10)

The second solution is negative and can be rejected.

The first solution is approximately 10.16 hours for A and 14.16 hours for B.

And people complain that they will never use algebra in real life!
Ah! I now see how I made a mistake. Your approach is correct.

t=4 + 2 sqrt(10) does equal 10.32 however, so the answer is 10.32 and 14.32 .
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Old 06-23-2014, 12:08 PM   #27
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Actually no, it does not make sense. I was working from the problem as originally presented in post #1 by NWB:
Without further ado, here's the puzzle.

Two persons, A and B, work together on a project and finish it in 6 hours. If working alone, A finishes it in 4 hours less than the time that B takes. How long does it take A and B to do it individually?
Oh, now I see how I made my mistake! I didn't read the question properly.
Hah! It's a good thing I refreshed before giving it another go, I missed your facepalm (I think that was edited?).

But now I'm curious where you went off? I was getting hung up for a while trying to keep it clear that the first case was a single project, and the second case is actually two projects being completed. It doesn't seem confusing looking back, but it was a little for me.

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Old 06-23-2014, 12:15 PM   #28
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Where I went wrong was in the interpretation of the word "individually". I was assuming that we were to figure out how much time each person contributed individually, while completing the task together as described in the first sentence. In retrospect, I now see that what NWB meant to convey was "if A and B were each assigned this task individually, how much time would it take each of them to complete it?".
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Old 06-23-2014, 12:26 PM   #29
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This reminds me about a math exam question given to my class in high school. The question had been lifted from a paper on a national exam for people several grades ahead, and our teacher had changed one word. That change completely altered the meaning of the question. Most people answered it "correctly" but silly me, I picked up on the word that had been changed, which made the proof much more difficult. (I don't remember the details but it had to do with the volume of a segment of a cone that was cut in a particular way). Anyhow, I got zero marks for my answer. My teacher would not listen to my complaint about the ambiguity of the revised question. I took it to my cousin who was a math teacher. Initially she sided with my teacher, but on reflection, she realized that the edit had made this a completely different problem. I never got the mark that I felt I had deserved, but at least I had some satisfaction. During my career I was always very careful to read the question. Obviously in ER I have let my guard down!
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Old 06-23-2014, 12:59 PM   #30
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t=4 + 2 sqrt(10) does equal 10.32 however, so the answer is 10.32 and 14.32 .
OK, I did not bother to check through Jjquantz's work to see that he had an error only when giving the numerical value at the very last step. Insignificant error. Grade changed to A-.

I found that piece of scratch paper I worked with yesterday. I solved for "speed of A", which turned out to be sqrt(5/72) - (1/6) = 0.096856471....

Invert the above numerical value, and we get 10.32455 hrs for time of A.
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Old 06-23-2014, 01:33 PM   #31
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I still think 4 & 8 are viable answers. Didn't anyone hear of negative workplace synergy? When we were very good at a task, much better than others, we could carry it out much faster alone, and the presence of a co-worker or manager caused productivity loss. How many times has anyone here said "I would get my work done much more quickly if they only would leave me alone". So the limitation that A>6 is a theoretical construct that doesn't necessarily apply to real life.

If B were a manager and A the direct report, there would be no doubt that A alone could do it in 4 but, working together with B, would take much longer.
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Old 06-23-2014, 01:56 PM   #32
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When I was at megacorp, they assigned me an underling so I could train him as a protégé. Now, I did not mind coaching a young engineer who showed enthusiasm about learning something. And I would love to mentor that super cute young woman engineer, but they did not assign her. Might as well, as I could get myself in trouble. But I digress.

If I let somebody do some of the work, while watching over his shoulders and coaching, it's gonna take more time than if I just do it myself. But they never gave me enough time, and kept hassling me about completion.
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Old 06-23-2014, 02:49 PM   #33
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In contrast, I like puzzles that have a precise answer. And that usually means math problems.

Though I like engineering, it always involves economic factors (else it would be called science). These bring fuzziness and a lot of debates into the solutions, but it is a fact of life that I have to accept.

Worse is when it involves politics, where the best technical solution is not adopted because it might upset someone in upper management, etc... It was one of the factors I ER'ed, although I have always liked my work and got paid quite decently.
Yes, it is nice to have a math problem to solve with a precise answer. So in real life one has to hunt high and low for something that is possible to abstract. That's why I chose the equity markets. However, real life problems do have a nasty habit of yielding somewhat fuzzy results. Heisenberg had something to say about that one didn't he.

I think maybe this is just me: don't try to find (and solve) hard problems when there are easier ones with a bigger payoff.

My hat is off to you guys that push right on into the hard stuff.
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Old 06-23-2014, 03:11 PM   #34
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I think maybe this is just me: don't try to find (and solve) hard problems when there are easier ones with a bigger payoff.

My hat is off to you guys that push right on into the hard stuff.
Are you kidding me? This is easy stuff, compared to deciding whether the market is efficient or not.

Look how quickly we dispose of this problem and reach a consensus, while we are debating forever whether we should buy or sell more stock. Man, that's really tough!

PS. I wonder what Marilyn thought about Market Efficiency Hypothesis. And more specifically, should we buy or should we sell now? I am still willing to listen to her. People are entitled to a goof once in a while.
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Old 06-24-2014, 08:00 AM   #35
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She was right if she had said in round numbers, how many hours, with 10 hours being the faster person and 14 hours the slower one....and the guy that used algebra was much closer, at 10.1 hours, but 10.3 hours and 14.3 hours are within 0.01 of being exact. i could not come up with a valid equation, since both workers contributions vary simultaneously, so i used trial and error. i knew the faster worker had to work slower than 6 hours to do the job alone, because both of them together took 6 hours to do it. that would mean B would not do any work....and i knew A had to work faster than 12 hours alone, because that would mean in 6 hours she would only do half the work when working with B and since he is 4 hours slower, he could not do the other half of the total during those 6 hours. so we know it takes A between 6 and 12 hours to do the job alone. if you plug in 10 hours for A, that means 14 for B. so in 6 hours, working at 10 hours for one, A would do 60%(6/10) of the total job. and B, in 6 hours, would do 0.4286(6/14) of the total...added together, that would be 1.0286 or 0.0286 more than the desired total. using that method, you come up with 10.31 hours being within 0.001 of the exact number...with a little more narrowing down, you could get even closer.
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Old 06-24-2014, 02:16 PM   #36
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... i could not come up with a valid equation, since both workers contributions vary simultaneously, so i used trial and error.

...with a little more narrowing down, you could get even closer.
In this case, one can come up with a closed-form solution (one that is mathematically definite and exact) as has been demonstrated in earlier posts. If that is too hard, or one does not know how, iterative solutions must be used.

Mathematicians always strive to get a closed-form solution, as they want exactness and do not like to guess. But in real life, many engineering problems are too complex to be solvable in closed-form. One then sets up the criteria that a potential solution has to satisfy, then tries different iterative methods to solve the problem numerically by writing a program running on a computer, hoping to converge on a solution. Before the invention of the digital computer, human calculators were used to crank out calculations by hand because they did not even have desk calculators.

See: Human computer - Wikipedia.

Potential problems with solving by numerical iteration are that a solution may not exist (computers run forever or the program crashes), slow convergence to a solution, or finding only one solution when multiple ones may exist.
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Old 06-24-2014, 02:59 PM   #37
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I still think 4 & 8 are viable answers. Didn't anyone hear of negative workplace synergy? When we were very good at a task, much better than others, we could carry it out much faster alone, and the presence of a co-worker or manager caused productivity loss. How many times has anyone here said "I would get my work done much more quickly if they only would leave me alone". So the limitation that A>6 is a theoretical construct that doesn't necessarily apply to real life.

If B were a manager and A the direct report, there would be no doubt that A alone could do it in 4 but, working together with B, would take much longer.
Exactly. Negative workplace synergy.

Person A would get the job done in 2 hours. Person B would get it done in 6. Person A having to work with Person B must deal with all the rabbit holes he wants to run down, and squelch all the good idea fairy tendencies, as well as deal with their questions and time-wasting practices. Frustrated, Person A says "Screw it," and goes and gets coffee while B fritters the time away.

Thus it takes them 6 hours - the same time it takes the weakest link to do it by himself.

This math has been tested and proven countless times in the real world. What do I win?
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Where Marilyn went wrong
Old 06-24-2014, 11:35 PM   #38
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Where Marilyn went wrong

I believed in Marilyn until I was able to see why she is wrong while trying to prove her right. Assuming that it took them the same total time working together as working separately will result in an incorrect answer in her solution. When they work together, the faster worker does more than half of the job and keeps the total time for two jobs lower. The slower worker does less than half of the two job total and has less influence on the total time. When working alone, each worker does exactly half of the total work of two jobs. The slower worker’s time to complete his half causes the two day total time to be greater than when they worked together doing the same total amount of work. The consecutive work time is not equal to the concurrent work time.
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Old 06-25-2014, 12:53 AM   #39
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This problem is deceptive because the nice rounded numbers lure people into trying to make clever reasoning and getting it wrong.

A and B working together can finish the job in 6 hrs. Individually, A takes 4 hrs less than B. How long does each take?


It appears that Marilyn's reasoning was the following.
If A+B finish 1 project in 6 hrs, then together they will finish 2 projects in 12 hrs.

Two persons working 12 hrs for 2 projects mean 24 workhours for 2 projects. She then assigns 10 out of the 24 hrs to A, and the remaining 14 hrs to B. This way A works 4 hrs less than B.

The above reasoning is wrong because 1 work-hour of A is worth more than 1 work-hour of B.

When verifying the answers, it is easily seen that when working together for 6 hrs, A will do 6/10 of the project, while B does 6/14 of the project. When summed up, 6/10 and 6/14 do not add up to 1.

As we have seen, the answers are 4 + 2*sqrt(10) and 8 + 2*sqrt(10).

Solving it a bit differently gives the 1st answer as 1 / ( sqrt(5/72) - 1/6) , which is the same as 4 + 2*sqrt(10).

Numerically, the time for A is 10.324553203... and for B is 14.324553203...

And 6/10.3245532 + 6/14.3245532 = 1!

All the above odd numbers are unexpected from the original innocent even numbers of 4 and 6 in the problem statement. Math problems can be deceptive!
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Old 06-25-2014, 01:03 AM   #40
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OK. How about 2 more different versions of the puzzle, so you can more readily see where a reasoning shortcut can lead you astray. People who do it the "hard way" should not bother, as they will get the right answer every time. However, people who apply shortcuts will get wrong answers that can be easily disproved, because the true answers will be nice rounded integers this time, I promise.


Case 2: A and B working together can finish the job in 2 hrs. Individually, A takes 3 hrs less than B. How long does each take individually?

Case 3: A and B working together can finish the job in 3 hrs. Individually, A takes 8 hrs less than B. How long does each take individually?
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