# Can you explain why Marilyn vos Savant was wrong?

Marilyn vos Savant writes the Ask Marilyn column in the Parade Magazine newspaper supplement. In her last column, she inadvertently created a nice little math problem. She did this by incorrectly answering a fairly non-interesting math problem. What is of interest is showing why an assumption she made was incorrect.

You can read the original problem here

This is my conversion of her original. Two people, a and b working together can complete a project in 6 hours. We know that if each of them works at the same rate as the other (assuming no synergy), it would take each of them 12 hours to complete the project on their own. In this case the sum of their work times is 24.

Suppose that a works faster than b. Marilyn assumed that the sum of their work times is still 24. Come up with an argument to show that the sum of their work times must in fact be greater than 24. You can in principle explain this with no algebra, but you might want to use algebraic notation to simplify the explanation.

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If you just want to know why she was wrong, it’s because she assumed that the relevant factor was the amount of time that had passed rather than the amount of work that was done.

If it takes them 6 hours to do one project together (12 work-hours total), it would take them 12 hours to do two projects together (24 work-hours total). But if this is their rate when together, then their rate when alone will be slower. Therefore, if it takes them 12 hours to two projects together (24 work-hours total), then it will take them more than 12 hours to do two projects separately (more than 24 work-hours total).

SavoirFaire (23996)

12 hours is still 12 hours, regardless of how much you accomplish in that time. So, the problem here is, are you talking work “hours” or the amount of work completed? Two different things.

I.E. Mary assembles 6 bicycles in 12 hours, Jane assembles 7 bicycles in 12 hours. How much work is completed? 13 bicycles. How many work hours – 24.

Skaggfacemutt (9785)

I don’t know… drag-asses always make the work take longer. The person working efficiently probably had to go back after everything was done and fix the lollygagger’s stupid mistakes, increasing the amount of time they had to spend working.

Seek (34714)

Because working separately, the entire process would take A’s time multiplicated with the factor by which A is faster than B.

Suppose A works twice as fast as B does. A would need 9h to finish the project on his own. By that time, B would still only be half-finished. A would have to wait around for another nine hours.

It seems clear to me, so I suspect I’m wrong.

longgone (12851)

@longgone , You are thinking along the right line.

Let me see if I can make what you are saying more specific. We know that A takes between 6 and 12 hours and that B takes more than 12 hours. Since A and B working together can complete a project in 6 hours, they can complete 2 projects in 12 hours. Imagine that they start working on separate projects. At some time x between 6 and 12 hours, A will have finished. The only way they can complete both projects in 12 hours is if A starts working together with B on B’s project for the remaining time. Can you complete the argument?

….because if A doesn’t work for part of the time, this time will get added to the total. We need both of them working non-stop for 24 hours to suffice. Am I right?

longgone (12851)

There is one key point that you are missing. Let x = time that A and B work separately. Let’s call the remaining time y. So we have x+y=12. When A and B work together for y hours, B does less than half the work. That means that when B works alone, the additional time to finish is greater than 2y. A takes x hours to complete the project and B takes more than x+2y hours, which means that the sum of their times is greater than 2x + 2y = 24.

Ah, got it. Thanks, I enjoyed trying to figure it out!

longgone (12851)

So you are assuming that the project has to be completed, not just that each works 12 hours, regardless of how much of the project they finish?

Skaggfacemutt (9785)

Yes, the assumption is that each person works separately until their project is complete.

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