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Math Help - Multiplying & Dividing Fractions - Application

  1. #1
    Junior Member Euclid Alexandria's Avatar
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    Question Multiplying & Dividing Fractions - Application

    From the book:
    "About 1/3 of all plant and animal species in the United States are at risk of becoming extinct. There are 20,439 known species of plants and animals in the United States. How many species are at risk of extinction? (Source: The Nature Conservancy)"

    Not so amusingly (being that I'm cramming), my solution results in 338,661 more species than there are known to be in the United States.

    The steps I took to solve the problem are illustrated in figure 1. The steps I took to list all the prime factors of the number 20,439 are illustrated in figure 2. I made my notes as clear as possible.

    Any hints regarding where I took a wrong step?
    Attached Thumbnails Attached Thumbnails Multiplying & Dividing Fractions - Application-multiplyingdividingfraction.jpg   Multiplying & Dividing Fractions - Application-multiplyingdividing.jpg  
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    Quote Originally Posted by Euclid Alexandria
    From the book:
    "About 1/3 of all plant and animal species in the United States are at risk of becoming extinct. There are 20,439 known species of plants and animals in the United States. How many species are at risk of extinction? (Source: The Nature Conservancy)"

    Not so amusingly (being that I'm cramming), my solution results in 338,661 more species than there are known to be in the United States.

    The steps I took to solve the problem are illustrated in figure 1. The steps I took to list all the prime factors of the number 20,439 are illustrated in figure 2. I made my notes as clear as possible.

    Any hints regarding where I took a wrong step?
    Hello,

    1. the prime factors of 20439 are: 20439=3^3 \cdot 757.
    That means 757 is a prime number and cann't be factorized any more.

    2. Therefore \frac{1}{3} \cdot 20439=\frac{1 \cdot 3 \cdot 3\cdot 3 \cdot 757}{3}=9 \cdot 757 = 6813

    Greetings

    EB
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    Quote Originally Posted by Euclid Alexandria
    ...
    The steps I took to list all the prime factors of the number 20,439 are illustrated in figure 2. I made my notes as clear as possible.

    Any hints regarding where I took a wrong step?
    Hello again,

    I've attached an image to show you, where you left the right path of truth .

    Greetings

    EB
    Attached Thumbnails Attached Thumbnails Multiplying & Dividing Fractions - Application-sum_prod.gif  
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    Junior Member Euclid Alexandria's Avatar
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    Question How do I tell if a very large number is a prime?

    That's not necessarily where I left the right path of truth. It's just an added mistake that I added rather than multiplying -- which wouldn't have worked either, of course.

    The place where I'm taking a wrong step is in not being able to determine whether very large numbers are prime numbers. When I pared 20,439 down to 20439=3^3 \cdot 757, I attempted to keep paring away. This was because I did not know how to determine whether 757 is a prime number.

    I know about certain tricks that can be used to divide numbers evenly, such as whether the sum of a number's digits is divisible by 9, or 3, or whether the last digit is divisible by 2, 5 or 10.

    Looking at 757, its last digit is a prime number, and the sum of its digits equals 19, also a prime number. Is this a number's way of saying, "I'm a prime?"
    Last edited by Euclid Alexandria; April 4th 2006 at 03:55 PM. Reason: Punctuation
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    Quote Originally Posted by Euclid Alexandria
    ....
    Looking at 757, its last digit is a prime number, and the sum of its digits equals 19, also a prime number. Is this a number's way of saying, "I'm a prime?"
    Hello,

    to factorize numbers is a tricky business:
    1. If the number isn't a square then you find a second factor if there is one factor; Example: 141 = 3 * 47.

    2. To find the first (the smaller) factor you only have to examine the number up to the \sqrt{number}

    3. With 757 you take all prime factors up to \sqrt{757} \approx 27.5

    4. a) 2, 3, 5 cann't be prime factors of 757 according to the rules you mentioned above.
    b)7 isn't a prime factor of 757, because 57 is not divisible by 7.
    c)11 isn't a prime factor of 757, because 13 is not divisible by 11.
    d)13 isn't a prime factor of 757, because 23 is not divisible by 13.
    e)17 isn't a prime factor of 757, because 77 is not divisible by 17.

    Do the same with 19 and 23. I'm quite sure, that you see how I did the factorization and how I know, that 757 is actually a prime number.

    5. I'll show you a few tricks to prove the divisibility(?):
    a) Divisible by 7: Take the hundreds of a number and multply them by 2. Add this result to the remainder of the number which is smaller then 100. If this sum is divisible by 7 then the original number is divisible by 7 too:
    259 / 7 ? :The hundreds: 2. So you calculate: 2*2+59=63. 63 is divisible by 7 so 259 is divisible by 7 too.

    1372 / 7?:13*2+72=98. 98 is divisible by 7, thus 1372 is divisible by 7

    b) Divisible by 11: Split the number into 2-digit-numbers, beginning at the end. Add those 2-digit-numbers. If this sum is divisible by 11, then the original number is divisible by 11 too:
    39248: 48+92+3=143, split again: 43+1=44 is divisible, so 39248 is divisible by 11 too.

    Greetings

    EB
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    Junior Member Euclid Alexandria's Avatar
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    Wait, wait....



    141 = 3 * 47? I am not seeing how that fits in with everything, or how you came up with it. I understood some of the things following that (and some other things I didn't). But to start with, could you please clarify that first point about 141?
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    Quote Originally Posted by Euclid Alexandria


    141 = 3 * 47? I am not seeing how that fits in with everything, or how you came up with it. I understood some of the things following that (and some other things I didn't). But to start with, could you please clarify that first point about 141?
    Hello,

    I gave you an example only. If you have a number and you want to get all prime factors you have to divide this number. The result of this division is greater then the divisor and this result contains(?) all the other factors which possibly exist.

    So take 156 = 2*78=2*2*39=2*2*3*13. If you know the smaller factors you know greater factors too. The greatest of the small factors never exceeds the value of \sqrt{156}\approx 12.48....

    In general: You only have to look for factors up to the sqrt(number). If you haven't found one up to this value then the number is a prime number itself.

    I hope that this will help a little bit. (My dictionary is nearly burning!)

    Greetings

    EB
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    Junior Member Euclid Alexandria's Avatar
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    Unhappy Thanks

    I wanted to come back to this post, and say thanks for attempting to help me out with this concept. I am still not grasping it, although I'm not sure why. Maybe if I come back to it later I'll be able to understand it more clearly.
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    Quote Originally Posted by Euclid Alexandria
    From the book:
    "About 1/3 of all plant and animal species in the United States are at risk of becoming extinct. There are 20,439 known species of plants and animals in the United States. How many species are at risk of extinction? (Source: The Nature Conservancy)"

    Not so amusingly (being that I'm cramming), my solution results in 338,661 more species than there are known to be in the United States.

    The steps I took to solve the problem are illustrated in figure 1. The steps I took to list all the prime factors of the number 20,439 are illustrated in figure 2. I made my notes as clear as possible.

    Any hints regarding where I took a wrong step?
    You write:

    <br />
x=\frac{1}{3}\ \times 20439=\frac{1}{3} \times \frac{20439}{1}=\frac{1 \times 20439}{3 \times 1}<br />

    but:

    <br />
\frac{1 \times 20439}{3 \times 1}=\frac{20439}{3}<br />

    so to find x you just divide 20439 by 3 (no prime factorisation needed)

    RonL
    Last edited by CaptainBlack; April 19th 2006 at 09:51 PM.
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  10. #10
    Junior Member Euclid Alexandria's Avatar
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    Post

    Thanks for the shortcut. The book asks that I follow prescribed steps, but I will remember this for checking my answers, and for when I'm in a hurry of course.

    Some of this stuff, I think, is just stuff that's eventually going to randomly click and make sense in my head. That's how it goes.
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