Class 9 Math Exercise 1.3 | Question 2

Question 2: You know that the value of \\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}), and \\frac{6}{7}) are, without doing the long division? If so, how?

Explanatory Answer | Exercise 1.3 Question 2

The decimal expansion of \\frac{1}{7} = 0.\overline{142857}). It is non-terminating recurring decimal.
Now, \\frac{2}{7}) = 2 × \\frac{1}{7}) = 2 × 0.\\overline{142857} = 0.\overline{285714})
\\frac{3}{7}) = 3 × \\frac{1}{7}) = 3 × 0.\\overline{142857} = 0.\overline{428571})
\\frac{4}{7}) = 4 × \\frac{1}{7}) = 4 × 0.\\overline{142857} = 0.\overline{571428})
\\frac{5}{7}) = 5 × \\frac{1}{7}) = 5 × 0.\\overline{142857} = 0.\overline{714285})
\\frac{6}{7}) = 6 × \\frac{1}{7}) = 6 × 0.\\overline{142857} = 0.\overline{857142})

Note that the same 6 digits appear in different order for each answer. Here is an easy way to remember these 6 values.
\\frac{1}{7} = 0.\overline{142857})
To find \\frac{2}{7}) multiply the first two digits of the decimal expansion of \\frac{1}{7}) with 2.
i.e., = 2 × 0.14 = 0.28; the decimal expansion of \\frac{2}{7}) will start with 0.28. Write the 6 digits starting from 28 in the order in which it appeared in \\frac{1}{7}). i.e., 0.\\overline{285714})
Along the same lines, if we have to quickly calculate the decimal expansion of \\frac{5}{7}) = 5 × 0.14 = 0.70. So, let us start from 7 and write the 6 digits. 0.\\overline{714285})