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Math Help - Percentage Problem: Understanding the Solution

  1. #1
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    Percentage Problem: Understanding the Solution

    Hi all,
    I was wondering if you could help me with the following...

    Q. The population of a certain town increases by 50 percent every 50 years. If the population in 1950 was 810, in what year was the population 160?

    Ans: 1650 , 1700, 1750, 1800, 1850.

    How I would have tackled this was to decrease 810 and the resulting amount by 50% and 50 years until I hit 160. But that's wrong. Here is the solution..

    Sol

    If the population increases by 50% every 50 years, the population in 1950 was 150% or 3/2 of the 1900 population. How; is it not just a 50% on 1900 so decreasing the value of 1950 by 50% should result in the 1900 figure?
    This mean the 1900 population was 2/3 that of the 1950.... and so on. How is that? 2/3 would be 66% would it not?

    A layman explanation so I can understand the principle here would be very much appreciated.

    Thanks in Advance,
    Paul
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  2. #2
    MHF Contributor Unknown008's Avatar
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    Ok, I'm not sure I fully understand the solution you provided, but this is how I think about it.

    Say in 1900, there was a certain population. That amount is 100%, you agree with me? Then, to get the population in 1950, you add 50% f that amount, or, you multiply by 150% (50% added to 100%).

    The reverse, is to divide by 150%, right?

    Now, each year, you will have to divide by 150%. This is a geometric progression and you can use the general formula:

    T_n = ar^{n-1}

    and convert it to:

    P_n = pr^{n-1}

    Where P_n is the population size after going back n times (that is, n = 50 years),
    p the initial population size, in this case 810.
    r is the common ratio, here it's 1/150%, or 1/(150/100) = 100/150 = 2/3\

    Here is where the 2/3 comes from.

    Then, at n times, the population became 160:

    160 = 810\left(\dfrac23\right)^{n-1}

    Solve for n.

    Then, multiply this by 50 years, subtract this from the year 1950.
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  3. #3
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    Thanks; so if the common ratio is 2/3, why is the first calculation performed using 3/2 do you know?

    Could I come to the same conculsion by reducing the amounts by 150% at a time?
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  4. #4
    MHF Contributor Unknown008's Avatar
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    Sure, it should bring you to the same result, because 2/3 is just a simplification of dividing by 150%
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  5. #5
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    Ok but is 150 not 3/2?
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  6. #6
    MHF Contributor Unknown008's Avatar
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    Yes, 150% = 150/100 = 3/2

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  7. #7
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    Quote Originally Posted by dumluck View Post
    Q. The population of a certain town increases by 50 percent every 50 years. If the population in 1950 was 810, in what year was the population 160?
    Ans: 1650 , 1700, 1750, 1800, 1850.
    Or look at it this way:
    160(1 + percentage_increase)^("n"umber of 50_year_periods) = 810
    160(1 + .5)^n = 810
    1.5^n = 810/160
    n = log(810/160) / log(1.5)
    n = 4

    SO: 1950 - 4*50 = 1950 - 200 = 1750
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  8. #8
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    Thanks. bare with me here. I'm sure we are on our way to a Eureka moment, so why are we multiplying each value by the reciprical? i.e. 2/3 if 150% = 3/2.
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  9. #9
    MHF Contributor Unknown008's Avatar
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    If we are going 'backwards', you do the reverse, and the reverse of multiplying by 150% is dividing by 150%.

    The fact that you have to divide several times by 150% might be tiresome, and what if you have to do that operation some dozen times! This is why you use the power.
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  10. #10
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    ahhh ok. So if the question asked for an increase in value we would be multiplying by 3/2?

    Good point. Thanks.
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  11. #11
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    Quote Originally Posted by dumluck View Post
    ahhh ok. So if the question asked for an increase in value we would be multiplying by 3/2?
    RIGHT! Same as 1.5 which I show in my example-solution ... see it?
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  12. #12
    MHF Contributor Unknown008's Avatar
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    Exactly right!
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  13. #13
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    Going back to the solution I showed you, you can "re-assure" yourself that 4 periods of 50 years is correct:

    160 * (1.5)^4 = 160 * 5.0625 = 810

    You can look at this as a financial transaction:
    if $160 is deposited and earns interest of 50% annually (yikes!), what will be the value in 4 years?
    F = A(1 + i)^n
    F = 160(1 + .50)^4
    F = 810 ... if no income tax!!
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  14. #14
    MHF Contributor Unknown008's Avatar
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    If interest was that much, banks would be ruined in no time while people would get rich extremely quickly
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