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Math Help - Fractions

  1. #1
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    Fractions

    Hi

    I spent a lot of time trying to sovle this so I would really appreciate it if someone could just give me some hints what I am doing wrong.

    Vicky.


    There are positive intergers k, n, m such that

    19/20 < 1/k + 1/n + 1/m < 1

    What is the smallest possible value of k+n+m?


    19 < 20 ( 1/k + 1/n + 1/m ) < 20

    19 < 20 ( nm + km + kn/ knm ) < 20

    nm + km + kn = try to make the value about 19.01


    knm = 20

    I did a lot of guessing and checking but now I am starting to see my solution doesn't make any sense..

    Vicky.

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  2. #2
    MHF Contributor Amer's Avatar
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    Quote Originally Posted by Vicky1997 View Post
    Hi

    I spent a lot of time trying to sovle this so I would really appreciate it if someone could just give me some hints what I am doing wrong.

    Vicky.


    There are positive intergers k, n, m such that

    19/20 < 1/k + 1/n + 1/m < 1

    What is the smallest possible value of k+n+m?


    just a try

    \frac{19}{20} < \frac{1}{k}+\frac{1}{n}+\frac{1}{m} < 1

    19 < \frac{20}{k} + \frac{20}{n} + \frac{20}{m} < 20

    take the numbers 2,3,6

    \frac{20}{2} + \frac{20}{3} + \frac{20}{6} = 10 + 6.6666 + 3.3333

    2+3+6 = 11 I think it is the smallest number
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  3. #3
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    Quote Originally Posted by Amer View Post
    \frac{20}{2} + \frac{20}{3} + \frac{20}{6} = 10 + 6.6666 + 3.3333

    2+3+6 = 11 I think it is the smallest number
    Not quite. 10 + 6.6666... + 3.3333... = 20 (we have repeating decimals), and the sum of the fractions has to be less than 20.


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  4. #4
    MHF Contributor Amer's Avatar
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    Quote Originally Posted by yeongil View Post
    Not quite. 10 + 6.6666... + 3.3333... = 20 (we have repeating decimals), and the sum of the fractions has to be less than 20.


    01
    the sum is less than 20

    10+3.3333333333... + 6.66666666... = 19.999999999999... < 20 right
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  5. #5
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    Quote Originally Posted by Amer View Post
    the sum is less than 20

    10+3.3333333333... + 6.66666666... = 19.999999999999... < 20 right
    No. Go back to the fraction form:
    \frac{20}{2} + \frac{20}{3} + \frac{20}{6}
    = \frac{60}{6} + \frac{40}{6} + \frac{20}{6}
    = \frac{120}{6}
    = 20

    Also, I believe it's been proven that 0.999.... (repeating forever) = 1.


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  6. #6
    MHF Contributor Amer's Avatar
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    Quote Originally Posted by yeongil View Post
    No. Go back to the fraction form:
    \frac{20}{2} + \frac{20}{3} + \frac{20}{6}
    = \frac{60}{6} + \frac{40}{6} + \frac{20}{6}
    = \frac{120}{6}
    = 20

    Also, I believe it's been proven that 0.999.... (repeating forever) = 1.


    01

    I think we should take 2,3

    ok take 2,3,7
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  7. #7
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    Quote Originally Posted by Amer View Post
    I think we should take 2,3

    ok take 2,3,7
    Again no, you get a fraction less than 19/20 (about 0.80952380952381).

    I'm starting to think that there is no solution.


    01
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  8. #8
    MHF Contributor Amer's Avatar
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    Quote Originally Posted by yeongil View Post
    Again no, you get a fraction less than 19/20 (about 0.80952380952381).

    I'm starting to think that there is no solution.


    01

    the answer is 19.5238095238095


    Fractions-coool.jpg
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  9. #9
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    I take that back. You're right -- 2, 3, and 7 does work. I was using a spreadsheet when figuring out 1/k + 1/n + 1/m and I must have inputted the formula wrong. My bad.


    01


    EDIT: I found out what I did. I plugged in 2, 6 & 7 by mistake. Oops!
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  10. #10
    MHF Contributor Amer's Avatar
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    Quote Originally Posted by yeongil View Post
    I take that back. You're right -- 2, 3, and 7 does work. I was using a spreadsheet when figuring out 1/k + 1/n + 1/m and I must have inputted the formula wrong. My bad.


    01


    EDIT: I found out what I did. I plugged in 2, 6 & 7 by mistake. Oops!

    no problem we all make mistakes thanks
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  11. #11
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    Thanks everyone for helping me out.

    Vicky
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  12. #12
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    Quote Originally Posted by Amer View Post
    just a try

    \frac{19}{20} < \frac{1}{k}+\frac{1}{n}+\frac{1}{m} < 1

    19 < \frac{20}{k} + \frac{20}{n} + \frac{20}{m} < 20

    take the numbers 2,3,6

    \frac{20}{2} + \frac{20}{3} + \frac{20}{6} = 10 + 6.6666 + 3.3333

    2+3+6 = 11 I think it is the smallest number
    Is this the equation I should have set up?
    And start plugging and checking the values?

    Vicky.
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  13. #13
    MHF Contributor Amer's Avatar
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    Quote Originally Posted by Vicky1997 View Post
    Is this the equation I should have set up?
    And start plugging and checking the values?

    Vicky.
    you can leave it but I prefer my equation

    19<\frac{20}{k}+\frac{20}{m}+\frac{20}{n}<20

    since the both right side and the left side are integers then I just start to guess numbers first take 2,2,2 that dose not work cuz the sum is 30

    it is clear that if we choose two numbers (2) the sum of them will be more than 20 so try

    2,3,3 dose not work too >20

    try
    2,3,4 dose not work too >20

    2,4,4 =20 the same

    3,3,3 < 19 so hold 2

    2,3,5 >20

    2,3,6 = 20 as you see

    r
    so
    2,3,7 work

    you can see that if we take bigger number the sum 20/k + 20/n + 20/m become smalle
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  14. #14
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    Thank you.

    Vicky.
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