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Ex.1.3 Q2 Number System Solution - NCERT Maths Class 9

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You know that \(\begin{align}\frac{1}{7} = \overline {0.142857} .\end{align}\) Can you predict what the decimal expansions of \(\begin{align}\frac{2}{7},\frac{3}{7},\frac{4}{7},\frac{3}{7},\frac{4}{7},\frac{5}{7},\frac{6}{7}\end{align}\) are, without actually doing the long division? If so, how?


 Video Solution
Number Systems
Ex 1.3 | Question 2

Text Solution

What is Known?

The decimal expansion of \(\begin{align}\,\frac{1}{7}\end{align}\).

What is Unknown?

The decimal expansions of \(\begin{align}\,\,\frac{2}{7},\frac{3}{7},\frac{4}{7},\frac{5}{7},\frac{6}{7}.\end{align}\)


\(\begin{align}\frac{1}{7}=0 . \overline{142587}\end{align}\)

This is a non-terminating recurring decimal.

We can use this to find the decimal expansion of \(\begin{align}\frac{2}{7},\,\,\frac{3}{7},\,\frac{4}{7},\,\frac{5}{7},\frac{6}{7}.\end{align}\)

To write the decimal expansion for:

i) \(\begin{align}\frac{2}{7}: \end{align}\) We observe that we get \(2\) as remainder after the second step in the above division.

 Hence,we start writing the quotient after the second decimal place and we get \(\begin{align}\frac{2}{7} = 0.\overline {285714} \end{align}\)

ii) \(\begin{align}\frac{3}{7}:\end{align}\) \(3\) is the remainder after the first step.

\(\begin{align}\rm{Hence}\; \frac{3}{7}=0.\overline {428571}\end{align}\)

iii) \(\begin{align}\frac{4}{7}:\end{align}\) \(4\) is the remainder at the \(4^{th}\) step.

 \(\begin{align}\text{Hence }\frac{4}{7}=0.\overline {571428} \end{align}\)

iv) \(\begin{align}\frac{5}{7}:\end{align}\)\(5\) is the remainder at the \(5^{th}\) step

\(\begin{align}\text{Hence }\frac{5}{7} = 0.\overline {714285} \end{align}\)

v)\(\begin{align}\frac{6}{7}:\end{align}\) \(6\) is the remainder after the \(3^{rd}\) step.

\(\begin{align}\text{Hence }\frac{6}{7} = 0.\overline {857142} \end{align}\)

 Video Solution
Number Systems
Ex 1.3 | Question 2
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