Goofs by a famous person

[NW-Bound]

This thread is spurred by a post on another thread here,

Quote:

Originally Posted by MichaelB No link or reproduction? How will we know?

where I reported that I found a mistake in a published puzzle by Marilyn vos Savant in Parade magazine yesterday, Sunday 6/22/2014.

For those who are not familiar with this, Marilyn is said to be the person with the highest IQ. I do not know how it can be measured in such absolute sense, but I do not doubt that she is among the top 0.1% or higher in intelligence, and always read her article with interest whenever I see it. I enjoy her opinions on questions on laws, philosophy, politics, etc... which are subjects that I am often not sure on. But she has also demonstrated logic and prowess in some scientific or math puzzles that stumped many learned readers. I still remember an incidence more than 10 years ago, because it caused such a stir at work that we discussed it at lunch, and agreed that Marilyn was right. She tore apart some readers who wrote in to say that she was wrong, and one of the readers was a professor. Ouch!

Anyway, when I read the puzzle yesterday, I thought to myself that this involved a bit of math, so I put the paper down and worked with a piece of scratch paper. Once I got my answers, I resumed reading to compare with Marilyn's answer, and what the heck!!!

She used some shortcuts, and I could not follow her reasoning at all, but could prove that my answers worked, but hers did not.

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?

Note that implying in the puzzle is that their collaboration has no synergy nor detraction, else the puzzle has too many unknowns. For example, if A can lay 100 bricks/hr, and B 50 bricks/hr, then working together they will lay 150 bricks/hr.

What's your answer?

PS. After reading yesterday's article, I searched the Web and found two more incidences where Marilyn was wrong in simular puzzles. They were in 2012 and 2013. Marilyn is 67 now, so I wonder if her mental acuity has been affected by age.

[Meadbh]

Let's assume that teamwork does not save A and B any time; their work is just additive. Between them, the job took 12 person-hours of work. Since there were two of them, we assume they did half the work each, so proportionately there would have been a two hour difference between them. Therefore

A + B = 12

And

A + 2 hours = B

So

A + A + 2 = 12

So

2A = 10

So

A = 5 hours and B = 7 hours

[NW-Bound]

Now, that we have more people getting interested and venturing a guess, here's a way you can double check your answer.

Suppose you guess, calculate, or miscalculate the answers as "4 and 8 hours". How do you verify the results? Simply by seeing if your answers fit the problem statements of course. Let's take the above "A does it in 4 hours and B does it in 8 hours" answers.

1) A takes 4 hours less than B. Checked.

2) Do they take 6 hours working together? No, not checked.

This is where most people have a problem with. I do not want to give out too much, but here's how you can find if your answers work: by using it in an example.

Let's say the work is laying 1000 bricks. In the answer above, A then lays 250 bricks/hr, while B does 125 bricks/hr. Together they will lay 375 bricks/hr. For 1000 bricks, it will take 1000/375 = 2.6666 hrs. It is not 6 hrs, so this answer set fails.

Note that we can tell that "4 and 8" fails right away, because if a single person does it in 4 hrs, then two of them together should not take 6!

=====>>>> Hence, A will have to take longer than 6 hours, and B will take exactly 4 hours more than that. <<<=======

I will now leave you to your calculator. This problem can be solved in a rather straightforward way with some equations, however. Shortcuts are dangerous!

[springnr]

To start with the time of day would have helped, siesta and such like could factor in, the key though is both A and B are good folk = until they are done.

[NW-Bound]

No work break allowed! A and B have to work straight through. Don't complicate matters furthermore here. You are eluding, I can tell.

[Yogi Bear]

So it seems working at the average speed (A+B)/2 the work takes 12 hours. (together they do it in 6 hours)

A would do the work 4 hours faster than B. Hence A=B-4.

Substituting in the equation above I get:

A=10
B=14

[NW-Bound]

Hi, Patient Bear! Welcome to the forum as I see that it is your 1st post.

Incidentally, your answer is what Marilyn came up with, but she used different wording!

You can check your answer using the example I set up in above post. It does not work!

A lays 1000 bricks/10 hrs = 100 bricks/hr, and B lays 1000 bricks/14 hrs = 71.428 bricks/hr.

So working together, they will take 1000/171.428 = 5.83 hrs. Close, but no cigars.

Eh, by now people should see how this can be solved.

[ERD50]

I also see the wording as too imprecise. There certainly could be synergy working together. In some other problem, you would be expected to 'discover' that as part of the solution.

It seemed pretty simple to me (but I do need paper and pencil, I can no longer visualize these problems in my head). But now, I think I over-simplified (and maybe that is where NW-B is going).

I didn't look at NW-Bs later post yet (trust me? - I just saw it pop up in preview, didn't read it).

I'm 99% sure that I agree she is wrong, but I'll post when I have the proof, and I give others a chance.

I'm also surprised she used the term 'man-hours' for Angelina?


possible partial spoiler alert - I'll put this in white text, drag your mouse over between my 'spoiler' tags to highlight to read it if you want .... if NW-B wants to tell me I'm on the right track (or not), that's OK

[spoiler] I think the mistake she makes is related to the 4 hours Brad works alone - in the baseline example, they are always working together, so Brad working alone for part of the time would take longer than the baseline 24 hours - but I need to map that out

[/spoiler]

-ERD50

[NW-Bound]

Again, in this kind of puzzle we do not want to complicate things with synergy or detraction or work break, etc... It's a simple math problem.

ERD50, no I have not posted my solution, but as I showed above, people venturing a guess or solution can verify their answer by double-checking against the problem statements.

[Lsbcal]

I wait with baited breath for your answer NW-Bound.

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.

[jjquantz]

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!

[ERD50]

OK, I have not looked at any other posts since #2, here's my proof that Marilyn is wrong, I have not calculated the correct answer yet though...


Like before, possible partial spoiler alert - I'll put this in white text, drag your mouse over between my 'spoiler' tags to highlight to read it if you want .... if NW-B wants to tell me I'm on the right track (or not), that's OK

[spoiler] I think the mistake she makes is related to the 4 hours Brad works alone - in the baseline example, they are always working together, so Brad working alone for part of the time would take longer than the baseline 24 hours -

The easiest thing I could think of is to back test to see if Marilyn is right/wrong. I approached this by assigning numbers to the 'project', assume the project is stacking X number of bricks. Since our key denominators in the answers for bricks/hour are 6, 12, 10 and 14, I factored those down to 3,6,5, and 7, and multiplied those to come up with 630 blocks in the project. Then all the answers are whole numbers (but you could use any number you wanted).

So if combined they complete a project in 6 hours, they stack an average of 630/6 hours = 105 blocks per hour. So we expect A to be faster than half that (>52.5), and B to be less than that.

Marilyn says A stacked these 630 blocks in 10 hours, that is 63 B/Hr.
Marilyn says B stacked these 630 blocks in 14 hours, that is 45 B/Hr.

Passes a common sense test so far, but...

OK, go back to them working together for 6 hours - that means:

A stacked 63 B/Hr for 6 hours = 378 blocks.
B stacked 45 B/Hr for 6 hours = 270 blocks.

So combined they would have stacked 648 blocks in 6 hours, and that does not match the 630 blocks that make up a project. So 10 and 14 is NOT the correct answer. There is another iteration required to determine their work rates or something. Sounds like you need to solve a simultaneous equation, yep, I bet that's it. Not too tough, but it will need to wait for later, while life intervenes.

[/spoiler]

-ERD50

[Bestwifeever]

The real answer is, if you are person A, you do not want to work with person B. The opposite is true if you are person B.

[NW-Bound]

And why does person B even have a job? Oh wait, both A & B work at a megacorp, so we know how that could happen. It should still be fair if A gets paid commensurately relative to B.

Wait a minute! Y'all keep eluding and try to distract me. My attention span is not that short yet.

Back to the problem. Your time is up.

[MichaelB]

Quote:

Originally Posted by NW-Bound And why does person B even have a job? Oh wait, both A & B work at a megacorp, so we know how that could happen. It should still be fair if A gets paid commensurately relative to B. Wait a minute! Y'all keep eluding and try to distract me. My attention span is not that short yet. Back to the problem. Your time is up.

Uh, no. In megacorp B is A's manager.

[Animorph]

The answer is A does it in 10.3246 hours, B does it in 14.3246 hours.

I had all the algebra for it, quadratic solution and all, but hit a key and erased it all accidentally.

I set it up using a = fraction of the job A completes per hour and b = fraction of the job B completes per hour and solved for a and b.

[ERD50]

Quote:

Originally Posted by Bestwifeever The real answer is, if you are person A, you do not want to work with person B. The opposite is true if you are person B.

Maybe B has a disability, and accommodations must be made according to the ADA. So it doesn't matter what A thinks, and she better not post any complaints to her Facebook page, or she is liable to lose her job, be sued, or both!

Now that I looked at and followed jjquantz solution, I'm thinking I would have approached it with simultaneous equations, but I am so, so, so very rusty that I'll need to look at that later.

-ERD50

[NW-Bound]

What I have seen is that if B is a manager, he/she would not have to do any work, and his/her time would be infinity. The problem is then ill-defined and has no solution. A will likely take early retirement, and now new employees C and D get hired to replace A. It is getting even messier.

Hence, let's assume that B is just a new and inexperienced employee, and not a manager.

[NW-Bound]

Animorph got it. Jjquantz's method is correct, but got an algebraic mistake somewhere.

Grades: A, and B+.

PS. I tend to make algebraic mistakes a lot more often now than when I was younger, hence my leniency.

PPS. Should I give the rest an "I" for incomplete? Well, no, take all the time you need. We are geezers and supposed to get slower after all. And we can beat Marilyn, even if for only this one time.

[Animorph]

Quote:

Originally Posted by NW-Bound Animorph got it. Jjquantz's method is correct, but got an algebraic mistake somewhere. Grades: A, and B+. PS. I tend to make algebraic mistakes a lot more often now than when I was younger, hence my leniency.

Yea!

Here's my equations:
a = fraction of job A completes per hour
b = fraction of job B competes per hour

6a + 6b = 1 (they compete 1 job in 6 hours)
1/b - 1/a = 4 (A takes 4 hours less to complete 1 job alone)