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

Tutorialspoint

Updated on: 10-Oct-2022

25 Views

Kickstart Your Career

Get certified by completing the course

Get Started