Goofs by a famous person

[Yogi Bear]

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 .

 

[ERD50]

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.

-ERD50

 

[Meadbh]

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?".

 

[Meadbh]

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!

 

[NW-Bound]

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.

 

[MichaelB]

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.

 

[NW-Bound]

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.

 

[Lsbcal]

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.

 

[NW-Bound]

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.

 

[hoosier daddy]

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.

 

[NW-Bound]

... 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.

 

[nash031]

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?

 

[Dave G]

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.

 

[NW-Bound]

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!

 

[NW-Bound]

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?​

 

[NW-Bound]

Before I forget, I must post this follow-up.

Marilyn publicly acknowledged her error in her column in Parade issue 7/13/2014, which is yesterday. She said that she was puzzled at first when people contacted her to say that she was wrong, but she finally saw the fault in her thinking.

Of course we have all seen people (including ourselves) being wrong in a logic or math problem that may be obvious to others (of course I am not talking about problems that are unprovable like EMH ). So, it's not schadenfreude but relief that I felt when I found that a smart person like Marilyn could be so wrong, and that so many people could spot her error like I did.

 

[NW-Bound]

I forgot to show answers to the variations I posed.

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?​

Case 2: A takes 3 hrs and B takes 6. To verify: 1/3 + 1/6 = 1/2.

Case 3: A takes 4 hrs and B takes 12. To verify: 1/4 + 1/12 = 1/3.

Nice numbers in this case. No crazy square roots like in the case that stumbled Marilyn.

 

[NW-Bound]

I look back at the above post, and see that there's an easier way to see that the answers work.

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?

Answer: 3 and 6 hours.

To verify, we can see that when working together for 2 hours, A will perform 2/3 of the job, and B 2/6 or 1/3 of the job. And 2/3 + 1/3 = 1 job.

 

[MikeD]

I get A = 1 hour and B = 5 hours.

Mike D.

 

[ERD50]

03 - AUG - 2014 column

While these may not be full on 'goofs', I really think someone of such high IQ should do a better job with educating people and using more precise language in her column. Three quibbles I have with today's column:

1) "Aluminum is the most abundant metal on the planet" - OK, that is true, but it's not the whole truth. Like hydrogen, almost all aluminum is bound up with other elements. We don't come across a 'vein' of aluminum like we do coal, gold, silver or lead, etc. It takes a large amount of energy to break that bond and make useful aluminum. That is why it is important to recycle it - melting it and reforming takes takes far less energy than refining it in the first place. So not a goof, but a lost opportunity, and maybe misleading to a lot of people.


2) "stainless steel conducts heat so unevenly" - makes me cringe. Stainless steel is a material, and conductivity is a property. For a given grade of stainless steel, its thermal conductivity is not uneven at all, its the same throughout.

It would be better to say "the temperature in a stainless steel pan can be uneven, due to the relatively poor conductivity of that metal" . Nitpicking maybe, but she is dumbing down her audience by using terms so loosely, IMO.


3) On using a magnet to distinguish an aluminum pot from a SS pot (some SS is somewhat magnetic, aluminum is not) - If a magnet doesn't stick "... you still don't know. The pot is either aluminum or it's not magnetized".

'Magnetized' is not the right term. 'Magnetic' is. 'Magnetic' means it can be 'magnetized', but it might not be. You can magnetize a steel screwdriver by swiping a magnet against it. You can demagnetize it in several ways. In either case, a magnet will stick to it.

I might let one nit-pick slide, but three in one short column is sloppy, and I don't think it reflects well on someone who makes a living based on publicizing her high IQ.

Anyone disagree? Did I get something wrong (never got my IQ tested, but I'm certain I'd test lower than Marilyn)?

edit/add: I see clintonwylie agrees with the last item ('magnetized') - no, I'm not him.

http://parade.condenast.com/323423/marilynvossavant/are-aluminum-pots-and-pans-harmful/

-ERD50

 

[ERD50]

10 - AUG - 2014 column - More multiple manglings

Sloppy, sloppy, sloppy! I really do expect more from such a bright person. She is giving 'High IQ' a bad name!

Today's questions was:

The railway industry claims that it achieves incredible gas mileage—about 450 miles per gallon. How is this possible?

Marilyn starts out OK (but not fully correct):

... railroad mileage is cited in ton-miles per gallon (instead of simply miles per gallon), it means something different. Rather, the term is more specific:

but then makes a slip that she should not have:

A freight train can move one ton of weight about 450 miles on a single gallon of gas.

Bzzzzzt! A freight train uses diesel, not 'gas'. Heck, even Elton John knows that ('get about as oiled as diesel train...')! You could say a gallon of 'fuel' or 'gasoline equivalent', but not 'gas'. Sloppy.

But it gets worse...

To match this mileage, a one-ton car would have to get 450 mpg, and a two-ton vehicle would have to get 225 mpg.

She is now off the tracks (keeping with the railroad theme). She got it mostly right above with 'ton-miles-per-gallon', but the real measure used in the example question is 'freight-ton-miles-per-gallon'. No one cares what the train weighs (or the truck/car for comparison) when we look at efficiency of moving freight, we care about the amount of freight it can move.

So a comparison with a one or two ton vehicle is meaningless - the real comparison is how much freight they carry, and their mpg to calculate an equivalent 'freight-ton-miles-per-gallon'.

Her explanation of the train efficiency isn't so hot:

Still, trains are extremely efficient. They have the benefit of an infrastructure that includes steel wheels rolling on steel rails (instead of soft rubber tires making frequent turns on pavement), highly sophisticated braking systems, and far, far fewer starts and stops.

Yes, rolling resistance is important. I'm not sure how 'highly sophisticated-braking systems' play into efficiency, AFAIK most freight trains do not have a storage batteries for regenerative braking (though they are hybrids of a sort - the diesel generator drives electric motors), so the brake energy is going to waste.

Fewer starts/stops, maybe. But a truck on the highway isn't stopping either, and it's highway 'freight-ton-miles-per-gallon' is still far lower at that point.

I'm pretty sure she is ignoring a big factor - wind resistance. For a train, the long line of cars are effectively 'drafting' each other.

I didn't quickly find any good comparisons, but this link has lots of interesting data:

http://www.istc.illinois.edu/about/SeminarPresentations/20091118.pdf

A truck's rolling resistance is 6-10x a trains, and they show that at 60 mph, the wind resistance of a train makes up about half of the total drag.

Lot's of other factors - just the size of a train brings 'economy of scale', you really don't need twice as much material to carry twice as much freight (the surface area of two one-gallon containers is greater than the surface area of one two-gallon container, etc).

And a BIG engine is more efficient (all things being equal) than a small engine - again, less surface area per HP to lose heat, etc. Probably less bearing surface area as well. Hmmm, but those big locomotives lose some eff% in converting mechanical to electrical to mechanical....

I expect better of her. I wonder if her mental acquity is diminishing with age, or some other factor. She really ought to do better than this.


-ERD50

 

[ERD50]

I did a little more looking into whether freight trains were starting to use regenerative braking, found some interesting stuff.

Turns out that a freight train that might have one or two locomotives for 100 cars or so, could not depend on braking only from the locos. All the cars have brakes. So to utilize regen, they would need motor/generators on every car. Since they don't stop that often, it sounds like it just does not make economic sense.

They are putting batteries into the small switching units in the yards. These sit and idle for long periods, then need bursts of energy to switch trains around, and start and stop. So it can work for that application.

I have considered that maybe commuter trains could take advantage of having an electric line installed for maybe 1/4 mile before and after stations. If they picked up the electric power for their motors directly from the line for acceleration out of the station, they would not burn so much diesel fuel. And if they injected the power back into the grid when they brake going into the station, that would essentially give them regenerative braking w/o having to carry batteries on the train. Even if they needed batteries to absorb and release the power surges that maybe the grid could not handle, they would be stationary batteries, which can be made much cheaper since they don't need to be small and light-weight. And they only need to store enough for that same train to leave the station.

Could I raise $1M by putting that on some crowdsourced funding site and finding a catchy phrase to put on coffee mugs and tee-shirts to give to 'donors'?

-ERD50

 

[ERD50]

02NOV2014 - Ooooopppps! She did it again!

Do Gifted Children Need Special Support? Plus, a Dandy Logic Puzzle

David Price in Decatur, Ill., writes:

If x and y are counting numbers (whole numbers above zero), what numbers do the terms (x÷2), (3÷y), and (y-x) represent?

Marilyn gives the answer as "1". But she makes the mistake of explaining this as the 'terms' needing to be 'counting numbers', but the problem only states that 'x and y are counting numbers'.

So there are infinite solutions. She has several comments to that effect already.

-ERD50

 

[NW-Bound]

The problem is poorly defined. It never states that (x÷2), (3÷y), and (y-x) are equal.

If (x÷2) = (3÷y) = (y-x), then solving for x and y we will get two solutions.

1) x = 2, and y = 3, then (x÷2) = (3÷y) = (y-x) = 1

2) x =-2, and y = -3, then (x÷2) = (3÷y) = (y-x) = - 1

If we now impose the condition that x and y are positive integers (counting numbers), then we retain only solution 1) above, and that's Marylyn's answer.

However, the problem statement never specifies that (x÷2), (3÷y) should be equal. Then, what does this mean? If x is a positive integer, then (x÷2) is either an integer or an odd multiple of (1/2). There is nothing else that we can conclude.

 

[ERD50]

I don't think she went down the path of thinking they would be equal - I think the mistake she made was thinking that the terms ((x÷2), (3÷y), and (y-x)) must be counting numbers, but the problem only states that x and y are counting numbers.

-ERD50