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Math Help - Word Problem/About squares

  1. #1
    Senior Member Mukilab's Avatar
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    Word Problem/About squares

    "In 1988 my Grannie was quite a bit older than Grandpa. In fact the difference between the squares of their ages (amazingly!) was exactly 1988.... How old were they?"

    My workings:
    x^2-y^2=1988

    (x+y)(x-y)=1988

    (x+y) or (x-y) must be even

    x or y must be even

    Now I'm stuck

    any help?
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  2. #2
    MHF Contributor
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    an (integer) solution is (78,64), but i could only get it with a spreadsheet. There are an unlimited number of solutions in real numbers that you could get from the quadratic formula
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  3. #3
    MHF Contributor

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    Quote Originally Posted by Mukilab View Post
    "In 1988 my Grannie was quite a bit older than Grandpa. In fact the difference between the squares of their ages (amazingly!) was exactly 1988.... How old were they?"

    My workings:
    x^2-y^2=1988

    (x+y)(x-y)=1988

    (x+y) or (x-y) must be even

    x or y must be even
    No, either x and y are both even or x and y are both odd.

    Now I'm stuck

    any help?
    I think the best way to do this is just by "brute strength".

    Lets just look at some possible numbers. Lets start With Grannie being, say 80. Then 80^2- x^2= 6400- x^2= 1988 so x^2= 4412. The positive root is 66.4.
    80^2- 66.4^2= 1991- a little too large. Grannie and Grandpa must be closer together in age.

    If Grannie were 75, we would have 75^2- x^2= 5625- x^2= 1988 so x^2= 3637 and Grandpa must be 60.3. Now, 75^2- 60.3^2= 1988.81.

    You can try some ages around 75 to see if you can come closer.
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  4. #4
    Member Veronica1999's Avatar
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    I found an easier way to solve it.
    If you look for the difference of two squares and make a pattern, you will see that it has to be in the form of
    4 (n+1) 3(2n+3)
    8(n+2) 5(2n+5)
    12(n+3) 7(2n+7)
    16(n+4) 9(2n+9)
    20(n+5)
    24(n+6)
    28(n+7)
    If you factor 1988 you get 4, 7, 71 , 28 times n+7 = 71 then n=64
    to get 28(n+7) means you are looking at the difference of n squared and (n+14) squared.
    Last edited by Veronica1999; June 28th 2010 at 06:56 AM.
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  5. #5
    Senior Member Mukilab's Avatar
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    Thanks Veronica for that different method but I still think brute force would be easier. Once I had eliminated the odd integers for x and y I went from 70 (since it's a grandmother) and did it in less than 20 seconds quickly substituting the integers on my calculator.

    Thanks for the help guys
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