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    When 1 is added to the numerator and denominator of the fraction m/n the new fraction is 3/2. When 1 is subtracted from the numerator and denominator of the fraction m²/n² the new fraction is 21/8. Find the possible values of m and n.

    I wrote this as two simultaneous equations:

    (m+1)/(n+1) = 3/2
    2(m+1) = 3(n+1)
    2m + 2 = 3n + 3
    2m = 3n + 1

    (2m)² = (3n + 1)²
    4m² = 9n² + 6n + 1
    8m² = 18n² + 12n + 2 <---(1)

    And the second equation:

    (m² - 1)/(n² - 1) = 21/8
    21/8 = (m² - 1)/(n² - 1)
    21(n² - 1) = 8(m² - 1)
    21n² - 21 = 8m² - 8 <---(2)

    Substitute (1) into (2):

    21n² - 21 = (18n² + 12n + 2) - 8
    21n² - 21 = 18n² + 12n - 6
    3n² - 12n - 15 = 0
    n² - 4n + 5 = 0

    And there are no answers for n.
    Can someone help me see where I've gone wrong please. Thanks.
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    Try factorising the second equation using the difference between two squares.
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    3n² - 12n - 15 = 0
    n² - 4n + 5 = 0
    you've divided the -15 by -3 for some reason

    so you'd have:

    n² - 4n - 5 = 0

    which is
    (n-5)(n+1)=0

    so n= 5 or n= -1
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    (Original post by SUKBarracuda)
    you've divided the -15 by -3 for some reason

    so you'd have:

    n² - 4n - 5 = 0

    which is
    (n-5)(n+1)=0

    so n= 5 or n= -1
    Thanks, that's (one) error, but still not the right answer.
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    (Original post by mik1a)
    When 1 is added to the numerator and denominator of the fraction m/n the new fraction is 3/2. When 1 is subtracted from the numerator and denominator of the fraction m²/n² the new fraction is 21/8. Find the possible values of m and n.

    I wrote this as two simultaneous equations:

    (m+1)/(n+1) = 3/2
    2(m+1) = 3(n+1)
    2m + 2 = 3n + 3
    2m = 3n + 1

    (2m)² = (3n + 1)²
    4m² = 9n² + 6n + 1
    8m² = 18n² + 12n + 2 <---(1)

    And the second equation:

    (m² - 1)/(n² - 1) = 21/8
    21/8 = (m² - 1)/(n² - 1)
    21(n² - 1) = 8(m² - 1)
    21n² - 21 = 8m² - 8 <---(2)

    Substitute (1) into (2):

    21n² - 21 = (18n² + 12n + 2) - 8
    21n² - 21 = 18n² + 12n - 6
    3n² - 12n - 15 = 0
    n² - 4n + 5 = 0

    And there are no answers for n.
    Can someone help me see where I've gone wrong please. Thanks.
    i.e 21(n+1)(n-1)=8(m+1)(m-1)
    so 21(n+1)(n-1)=8((3n+1)/2+1)((3n+1)/2-1)
    so 21(n+1)(n-1)=(12n+12)(3n/2-1/2)
    =18n^2+12n-6
    so 3n^2-12n-15=0
    so 3(n^2-4n-5)=0
    so 3(n-5)(n+1)=0
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    n=5 is correct but n cannot be -1
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    (Original post by hornblower)
    Try factorising the second equation using the difference between two squares.
    How does that help? It just gives a more complicated equation to simplify.
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    (Original post by lgs98jonee)

    so 3n^2-12n-15=0
    so 3(n^2-4n-5)=0
    so 3(n-5)(n+1)=0
    The top line is what I got, but that isn't the right answer.
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    (Original post by mik1a)
    The top line is what I got, but that isn't the right answer.
    when n=5,m=8 and so (8+1)/(5+1)=9/6=3/2
    and (8^2-1)/(5^2-1)=63/24=21/8.
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    (Original post by IntegralAnomaly)
    when n=5,m=8 and so (8+1)/(5+1)=9/6=3/2
    and (8^2-1)/(5^2-1)=63/24=21/8.
    But I can't guess then in an exam and prove they are right. Does anyone have a proper working?
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    (Original post by mik1a)
    But I can't guess then in an exam and prove they are right. Does anyone have a proper working?
    guess? u worked it out earlier! n^2-4n-5=0
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    (Original post by IntegralAnomaly)
    guess? u worked it out earlier! n^2-4n-5=0
    Oh yeah duh. THanks!

    I thought that the two values of n were one value of n and one of m! You can make some really stupid assumptions when you're not concentrating on it!

    Thanks everyone! (good start to P2!)
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    (m+1)/(n+1) = 3/2
    2(m+1) = 3(n+1)
    2m + 2 = 3n + 3
    2m = 3n + 1

    (2m)² = (3n + 1)²
    4m² = 9n² + 6n + 1
    8m² = 18n² + 12n + 2 <---(1)

    And the second equation:

    (m² - 1)/(n² - 1) = 21/8
    21/8 = (m² - 1)/(n² - 1)
    21(n² - 1) = 8(m² - 1)
    21n² - 21 = 8m² - 8 <---(2)

    Substitute (1) into (2):

    21n² - 21 = (18n² + 12n + 2) - 8
    21n² - 21 = 18n² + 12n - 6
    3n² - 12n - 15 = 0
    n² - 4n - 5 = 0
    (n-5)(n+1)

    so n=5 or -1

    but can't be -1

    so n=5

    therefore m = 8.

    Your working was all right except for that one mistake...
 
 
 
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