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Math Help - Scale problem

  1. #1
    Member GAdams's Avatar
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    Scale problem

    When you cut a piece of A4 paper in half (portraiot way up, cut across) youget two pieces of A5 paper.

    Ignoring its thickness, A5 paper is similar to A4 paper.

    Express the length of A5 paper as a percentage of the length of A4 paper.



    OK, I get that A5 is 1/2 of A4. So, that would make it 50% of A4. But this is wrong.


    Could someone please explain this step by step please?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by GAdams View Post
    When you cut a piece of A4 paper in half (portraiot way up, cut across) youget two pieces of A5 paper.

    Ignoring its thickness, A5 paper is similar to A4 paper.

    Express the length of A5 paper as a percentage of the length of A4 paper.



    OK, I get that A5 is 1/2 of A4. So, that would make it 50% of A4. But this is wrong.


    Could someone please explain this step by step please?
    As you get two A5 pieces from an A4 sheet, the area of an A5 is half
    that of A4. The areas scale as the square of the linear dimensions, so
    the length of an length(A5) sheet is length(A4)/sqrt(2).

    RonL
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  3. #3
    Member GAdams's Avatar
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    OK. So if the area is 1/2, then the length will be the square root of 1/2.

    Which is 0.707

    If I multiply this by 100, I should get the length of A5 as a percentage of A4

    Which is rounded to 71%.
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by GAdams View Post
    OK. So if the area is 1/2, then the length will be the square root of 1/2.

    Which is 0.707

    If I multiply this by 100, I should get the length of A5 as a percentage of A4

    Which is rounded to 71%.
    Yes. And the great secret of many problems like this is you can actualy
    measure a peice of A4. The width will be the height of the A5 and its height
    is of course the height of a piece of A4.

    RonL
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  5. #5
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    Hello, GAdams!

    When you cut a piece of A4 paper in half (portrait, cut across)
    you get two pieces of A5 paper.

    A5 paper is similar to A4 paper.

    Express the length of A5 paper as a percentage of the length of A4 paper.

    Another approach (to me, the obvious one) . . . make a sketch!
    Code:
        - *-----------* -
        : |           | :
        : |           | ŻL
        : |           | :
        L * - - - - - * -
        : |           | :
        : |           | ŻL
        : |           | :
        - *-----------* -
          :     W     :

    A4 has length L, width W.

    A5 has length W, width \frac{1}{2}L.

    We want the ratio: . \frac{W}{L}


    Since the rectangles are similar: . \frac{W}{\frac{1}{2}L} \:=\:\frac{L}{W}\quad\Rightarrow\quad W^2\:=\:\frac{1}{2}L^2\quad\Rightarrow\quad\frac{W  ^2}{L^2} \:=\:\frac{1}{2}<br />

    Hence: . \left(\frac{W}{L}\right)^2 \:=\:\frac{1}{2}\quad\Rightarrow\quad \frac{W}{L} \:=\:\frac{1}{\sqrt{2}} \:=\:0.707106781


    Therefore: W is about 70.7% of L.

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