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Calling for help

AhQ ['a: kiu]
User since 2006
Posts: 2715

3 April 2011

I am living in a small flat with used furniture, rats and mice. The walls are dirty pink. Two friends of mine, William and Serena, assured me they could turn my dirty room into a nice and healthy place to live in. They would repaint the walls with a stuff especially made to get rid of rats and mice. Serena said she could get it over with the job in a 2 days time. William, her lame buddy, should take 6 days for the same workout. But they decided to work together starting the job on this Monday the 4th. The problem is that I am not able to calculate the day when I may move back to my flat, knowing that during the paint works I am bound to live as a homeless, sleeping under bridges.
Is there here around some kind GGuser at the ready to tell me when I can move back home? Many thanks in advance.
Joe

11 comments

  • User since 2008
    Posts: 499

    4 April 2011

    An interesting, challenging conundrum. This chore should be something for Metrics Controllers. He likes statistics and counting. But I like those kind of things.


    I would be a precipitous conclusion that in 2 days, William would have done a third of Serena's work - and hence would have saved her 2/3 days – and the work is done in 1+1/3 day. No.


    The precise time (in the mathematical sense) can't be calculated accurately- provided we presuppose that both finish their work exactly at the same instant and nobody waits for other other to finish. Instead, it's an approximation, a series (something like a Taylor Series) – a series
    always converges to a value.


    The deliberation to the solution is the following:


    At the beginning, each of both is assigned 1/2 of the work. After one day, Serena's finishes her part, and by that time, William has only finished a third of hers (1/6). The remaining work is 1/3. Then, Serena takes over again the half of remaining work (1/6) . She finishes that part in again 1/3 day, and William has only done 1/18. Again, Serena takes over half of the work - and so on.


    So, to put it more schematically:
    -In 1 day, Serena finishes her half and 1/3 of the work remains. Serena
    takes over 1/6.
    -In another 1/3 day, Serena finishes her half, and 1/9 of the work remains, Serena takes over 1/18.
    -In another 1/9 day, Serena finishes her half, and 1/27 of the work remains. Serena takes over 1/54


    and so on.


    If you calculate that up to, say, 6 steps, a pattern emerges, which gives the hint to the corresponding series:


    The resulting series is:


    Days until work is finished: 1 + 1/3 + 1/9 + 1/27 + 1/81 + 1/243 + … +
    In mathematical terms: Sum (from n=0 to n=infinite) of: (1/ (3 to the n-th))


    That is, the sum converges to a value (and will never reach it).


    In order to get a image how long it takes until you can return to your refurbished home, let's assume 9 steps:


    For n = 8 (Serena and William split their work 9 times), the approximation is 1,49999237923 days.


    That is:


    1 day, 11 hours, 59 minutes, 53 seconds, 4156 milliseconds and 288 microseconds. That should be precise enough for you get back home in time.


    Cordially
    Whacky
  • User since 2008
    Posts: 253

    4 April 2011

    Joe did not say how many hours per day his friends will work ?
    Whacky, i think you should recalculate the time according to differents working hours ?
    and if they work" non stop", perhaps Joe won't have to sleep outside, not even once....better for his health,
    it's still rather chilly these days.
    By the way : which colour ?
    cheers
    Victorine
  • User since 2008
    Posts: 499

    4 April 2011

    The 'real time* in full days, hours, minutes and seconds depends on how many hours a working day of Serena and William consists of.


    The factor 1,4999... is independent of the number of hours in a working day.


    Nevertheless, the last calculation is (as you might have noticed) based on the proviso that his friends work in 24-hour shifts (non-stop that is). I have an overstated, maybe unrealistic faith in the motivation of craftsmen - and that his best friends will do everything for him - and get the work done as quick as possible - for Joe's health sake.


    'That's what friends are for...'.


    If I deduced this lowbrow formula wrongly (did I or did I not?), then Joe has to take his cellphone with him and wait for his friends to call him via SMS when the work is done. The live of a 'clochard' has no feeling of time. Even in an April Fool or in a contrived story as math text problem.
    Cheerio
    Whacky
  • User since 2006
    Posts: 2715

    5 April 2011

    So, this April 4 at dawn I left my flat as Serena and William dawned at the door. Wandering about as a lost soul, I came across the Metrics Controller who told me he had bad news. On his laptop screen he showed me the skilful calculation the top-mathematician of the forum had cordially worked out of pity for me. Mathematically speaking, Serena and William couldn't reach the end of the job! I was taken aback because I didn't catch how an irrational number could ever look like a Danaides' jar. Anyhow I realized that I was bound to sleep under bridges for the rest of my life. (I thought to myself: Let's hope that won't last long.)
    Fortunately Whacky is a candid soul always at the ready to help. He suggested a cellphone could possibly be useful for someone thrown in the streets.
    Victorine also wanted to console me. She insinuated that, if I didn't know the colour of the painting, it wasn't worth moving back.
    Now then, I want to thank both of them, Whacky and Victorine, for being so much attentive as to give a follow-up to my request.
    Joe, a new one among the homeless.
  • User since 2006
    Posts: 2715

    5 April 2011

    Don't worry, Joe. I can host you. Just knock at my door, fifth landing, third door on the right.
    My name is on the door, 'Ayeomen'.
  • User since 2008
    Posts: 499

    6 April 2011

    Mathematical speaking, the complete work will always be less than 1,5 working days. The formula is a geometrical series, and the sum formula for this series shows that for an infinite number of splits, the time will converge to 1.5 working days.


    That means the Serena and William always need less than 1.5 working days, never exactly or more than that for their job. That is the result of the whole calculation.


    Sorry for bothering all of you with this grisly mathematical stuff. I know it's boring and even repulsive tor many people. But I took the opportunity to ponder about this mathematical text problem. It was fun.
  • User since 2008
    Posts: 499

    6 April 2011

    Revised version:


    Mathematically speaking, the complete work will always amount to less than 1,5 working days. The formula is a geometrical series, and the sum formula for this series shows that for an infinite number of splits, the time will converge to 1.5 working days.


    That means that Serena and William will always need less than 1.5 working days, never exactly or more than that for their job. That is the result of the whole calculation.


    All that aside, sorry for bothering you all with this grisly mathematical stuff. I know mathematics is boring and even repulsive tor many people. Nevertheless, I took the opportunity to ponder about this mathematical text problem and to brush up on my rusty maths. As far as I can recall, series were a subject of 11th grade. Back then, I didn't like series at all - to say the least.
  • User since 2006
    Posts: 2715

    6 April 2011

    Thank you, Whacky, for your lesson about series. As I was plain stupid I figured out that as 1,4999999999... going so on with googols of googols of 9 up towards the infinite woud never come to an end I would never see my flat again.
    But anyway there were two good omens:
    Primo, you assured that the couple of workers would need less than 1.5 day to get over with the job.
    Secundo, Sandy was about to host me. She has a heart of gold.
    So I didn't sleep under a bridge more than one night. Good to me, for it was a bit chilly. Vic had warned me.
  • User since 2008
    Posts: 499

    6 April 2011

    From Joe the screwball:
    Thank you, Whacky, for your lesson about series. As I was plain stupid I figured out that as 1,4999999999... going so on with googols of googols of 9 up towards the infinite woud never come to an end I would never see my flat again.
    But anyway there were two good omens:
    Primo, you assured that the couple of workers would need less than 1.5 day to get over with the job.
    Secundo, Sandy was about to host me. She has a heart of gold.
    So I didn't sleep under a bridge more than one night. Good to me, for it was a bit chilly. Vic had warned me.

     


    Exactly. You're welcome. Fun while it lasted. It's been a pleasure. It's been a treat and so on.

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