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You know that \( \frac{1}{7}=0 . \overline{142857} \). Can you predict what the decimal expansions of \( \frac{2}{7}, \frac{3}{7} \). \( \frac{4}{7}, \frac{5}{7}, \frac{6}{7} \) are, without actually doing the long division? If so, how?
Given:
\( \frac{1}{7}=0 . \overline{142857} \)
To do:
We have to predict the decimal expansions of \( \frac{2}{7}, \frac{3}{7}, \frac{4}{7}, \frac{5}{7}, \frac{6}{7} \) without actually doing the long division.
Solution:
\( \frac{1}{7}=0 . \overline{142857} \)
Therefore,
\( \frac{2}{7}=2\times\frac{1}{7}=2\times0 . \overline{142857}=0.\overline{285714} \)
\( \frac{3}{7}=3\times\frac{1}{7}=3\times0 . \overline{142857}=0.\overline{428571} \)
\( \frac{4}{7}=4\times\frac{1}{7}=4\times0 . \overline{142857}=0.\overline{571428} \)
\( \frac{5}{7}=5\times\frac{1}{7}=5\times0 . \overline{142857}=0.\overline{714285} \)
\( \frac{6}{7}=6\times\frac{1}{7}=6\times0 . \overline{142857}=0.\overline{857142} \)
Another way to do it is studying the remainders while finding the value of $\frac{1}{7}$.
7)1.0(142857
7
-------
30 $\Longrightarrow$ $\frac{3}{7}=0.\overline{428571}$
28
---------
20 $\Longrightarrow$ $\frac{2}{7}=0.\overline{285714}$
14
-----------
60 $\Longrightarrow$ $\frac{6}{7}=0.\overline{857142}$
56
-----------
40 $\Longrightarrow$ $\frac{4}{7}=0.\overline{571428}$
35
------------
50 $\Longrightarrow$ $\frac{5}{7}=0.\overline{714285}$
49
--------------
1