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Arithmetic progression

  1. Jul 10, 2011 #1
    1. The problem statement, all variables and given/known data
    A woman started a business with a workforce of 50 people. Every two weeks the number of people in the workforce increased by 3 people. How many people were there in the workforce after 26 weeks?
    Each member of the workforce earned $600 per week. What was the total wage bill for this 26 weeks?


    2. Relevant equations
    Un = a + (n - 1) d
    Sn = n/2 (a + Un)


    3. The attempt at a solution
    Number of terms = 26 / 2 = 13
    Number of people = 50 + (13 - 1) x 3 = 86 --> wrong :grumpy: The answer is 89. I don't understand

    first week = 50 people
    second week = 50 people
    third week = 53 people
    and so on...
    Then, total wage bill = 13/2 (50 + 89) x 2 x 600 = 1 084 200 ---> wrong. :cry: The answer is 1 341 600

    Please help me. Thanks
     
  2. jcsd
  3. Jul 10, 2011 #2

    tiny-tim

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    hi songoku! :wink:
    why (13 - 1) ? :confused:

    she started with 50

    after 2 weeks, 50 + 1*3 = 53

    after 26 weeks, 50 + 13*3 = 89 :smile:
     
  4. Jul 10, 2011 #3
    hi tiny-tim :wink:

    Because every two weeks the number of people in the workforce increased and there are 26 weeks, so I thought the number of terms = 26 / 2 = 13. Then by using Un = a + (n - 1) d, I got 86. Why can I do it this way?
     
  5. Jul 10, 2011 #4

    tiny-tim

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    hi songoku! :wink:

    never mind the formula,

    you're not reading the English properly :redface:

    it clearly says that she starts with 50, and only gets another 3 at the end of the first 2 weeks …

    that's 14 terms! :smile:
     
  6. Jul 10, 2011 #5
    hi tiny-tim :wink:
    oh yes yes yes. I misinterpreted the question. But I still don't know how to find the total bill. Thanks
     
  7. Jul 10, 2011 #6

    SteamKing

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    Since you miscalculated the total workforce after 26 weeks, don't you think you should re-calculate the total wages, now that you know how many people are working at a given time?

    All too often, students give up after the first try at something.
     
  8. Jul 11, 2011 #7
    I am not so sure what you mean but if you look at my first post, you would see that I used 89 workforce instead of 86. I already used the correct answer for calculating the total wage bill and still got the wrong answer.

    And in my calculation, I used 13 as number of terms, not 14 because I thought at the beginning, the workforce didn't work so no wage spent. The wage starts from first week until 26th week so there is 26 terms. I have written my calculation in first post. Please show me my mistake.

    Thanks
     
  9. Jul 12, 2011 #8

    tiny-tim

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    hi songoku! :smile:
    this formula is wrong …

    you need the average of the original amount (50) and the final amount (50 + …)

    so you shouldn't halve the 50 :redface:
     
  10. Jul 15, 2011 #9
    hi tiny-tim :smile:

    I don't understand what you meant...:redface:

    This is my thought:
    number of workforces from 1st week until 26th week:
    50 50 53 53 56 56 59 59 62 62 65 65 68 68 71 71 74 74 77 77 80 80 83 83 86 86

    eh, why did I get 86 instead of 89 workforces at the end of 26th week?? I think I made mistake here but I couldn't figure it out....let me think about it for a while

    Thanks
     
  11. Jul 15, 2011 #10

    NascentOxygen

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    Hello songoku. The problem is not well worded, so it's not all your fault that you are not getting the "right" answer. :smile: It relies on a tiny nuance in the language.

    It would be better if the teacher had written the problem as "during every two week period" the number of employees increases by 3. That way, it would be clear that by the end of the first two weeks there will be 53 employees.
    It is a really good idea to write things out fully, like you have here, because it allows you to compare reality with the mathematics, and determine where the fault lies.
     
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