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ioana12
Pe: 2007-11-02T19:26:14+02:00 In: In:

problema clasa a vi-a

rezolvati in N ecuatiile :

_____ ______ ______
0,(1x)+0,(2x) +…+0,(9x) = x si

_____ ______ ______
0,(x1)+0,(x2) +…+0,(x9) = x , unde numerele sunt scrise in baza 10.
m-am blocat ! va multumesc mult

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5 raspunsuri

  1. dan
    expert (VI)

    ioana12 wrote: rezolvati in N ecuatiile :
    ____ ______ ______
    0,(1x)+0,(2x) +…+0,(9x) = x si

    _____ ______ ______
    0,(x1)+0,(x2) +…+0,(x9) = x , unde numerele sunt scrise in baza 10.
    m-am blocat ! va multumesc mult


    \overline{0,(1x)}=\frac{\overline{1x}}{99}=\frac{10+x}{99};\\ \overline{0,(2x)}=\frac{\overline{2x}}{99} =\frac{20+x}{99};\\ .........................................................................\\ \overline{0,(9x)}=\frac{\overline{9x}}{99}=\frac{{90+x}}{99};

    Facem insumarea pe gupe, termen cu terme si obtinem:\\
    \sum=\frac{10}{99}+\frac{20}{99}+...+\frac{90}{99}+\frac{x}{99}+\frac{x}{99}+...+\frac{x}{99}= \frac{10\cdot(1+2+...+9)}{99}+\frac{9x}{99}=\frac{10*9*10:2}{99}+\frac{9x}{99}=\frac{50}{11}+\frac{x}{11}= \frac{50+x}{11}=x
    de unde, obtinem 50+x=11x => x=5

    Asemanator procedam si la exercitiul 2)

  2. Avram_Celina

    \begin{array}{l} \overline {0,(1x)} + \overline {0,(2x)} + \overline {0,(3x)} + ... + \overline {0,(9x)} = \frac{{\overline {1x} }}{{99}} + \frac{{\overline {2x} }}{{99}} + \frac{{\overline {3x} }}{{99}} + ... + \frac{{\overline {9x} }}{{99}} \\ = \frac{{10 + x + 20 + x + 30 + x + ... + 90 + x}}{{99}}(scrierea - in - baza - 10) \\ = \frac{{9x + 10 + 20 + 30 + ... + 90}}{{99}} \\ = \frac{{9x + 10(1 + 2 + 3 + ... + 9)}}{{99}} \\ = \frac{{9x + 10*\frac{{9*10}}{2}}}{{99}} \\ = \frac{{9x + 10*9*5}}{{99}} = x \\ \Rightarrow 9x + 450 = 99x \\ \Rightarrow 450 = 90x \Rightarrow x = 5 \\ \end{array}

  4. ioana12

    la indicatii zice x=5 …

    multumesc dan !

  5. ioana12

    gata am ghicit-o !!! merci mult

  6. ioana12

    10 + 20 + …+ 90 + x + x + x +… x
    – – – – – – –
    99 99 99 99 99 99 99
    10(1+2+…+9) + 9x = 10*9*10:2 + 9x =
    –––––– – –––– –
    99 99 99 99

    10*45 + 9x = 450 + 9x = 50+x = x
    ––- – –- – ––
    99 99 99 99 11

    50+x=11x
    50=10x
    x=5

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