You know that 1/7 = I.142857 bar. Can you predict what the decimal expansions of 2/7, 3/7, 4/7, 5/7, 6/7 are, without actually doing the long division? If so, HOWAND WHY?

Asked by Shashi Bushan | 10th Mar, 2017, 06:58: PM

Expert Answer:

begin mathsize 16px style Observe space the space remainders space while space finding space the space value space of space 1 over 7.
Given space that space 1 over 7 equals 0.142857 with bar on top
2 over 7 equals 2 open parentheses 1 over 7 close parentheses equals 2 open parentheses 0.142857 with blank on top close parentheses equals 0.285714
3 over 7 equals 3 open parentheses 1 over 7 close parentheses equals 3 open parentheses 0.142857 with blank on top close parentheses equals 0.428571
4 over 7 equals 4 open parentheses 1 over 7 close parentheses equals 4 open parentheses 0.142857 with blank on top close parentheses equals 0.571428
5 over 7 equals 5 open parentheses 1 over 7 close parentheses equals 5 open parentheses 0.142857 with blank on top close parentheses equals 0.714285
6 over 7 equals 6 open parentheses 1 over 7 close parentheses equals 6 open parentheses 0.142857 with blank on top close parentheses equals 0.857142
end style
This can be done using multiplication. But there is no need to do that even. Directly you can observe the decimal part of 1/7.
To multiply 2(1/7), we shift the decimal two places to the right. 
Similarly, to multiply 2(1/7), we shift the decimal point, three places to the right.
The same can be done for the others.
This happens since 1/7 is recurring and hence you will get the same remainders again and again after a certain pattern.

Answered by Rebecca Fernandes | 10th Mar, 2017, 09:03: PM