You know that 17\dfrac{1}{7}71 = 0.142857‾0.\overline{142857}0.142857. Can you predict what the decimal expansions of 27\dfrac{2}{7}72, 37\dfrac{3}{7}73, 47\dfrac{4}{7}74, 57\dfrac{5}{7}75, 67\dfrac{6}{7}76 are, without actually doing the long division? If so, how?
[Hint : Study the remainders while finding the value of 17\dfrac{1}{7}71 carefully.]
Given,
17\dfrac{1}{7}71 = 0.142857‾0.\overline{142857}0.142857
(i) 27\dfrac{2}{7}72 = 2×172 \times \dfrac{1}{7}2×71
= 2×0.142857‾2 \times 0.\overline{142857}2×0.142857
= 0.285714‾0.\overline{285714}0.285714
(ii) 37\dfrac{3}{7}73 = 3×173 \times \dfrac{1}{7}3×71
= 3×0.142857‾3 \times 0.\overline{142857}3×0.142857
= 0.428571‾0.\overline{428571}0.428571
(iii) 47\dfrac{4}{7}74 = 4×174 \times \dfrac{1}{7}4×71
= 4×0.142857‾4 \times 0.\overline{142857}4×0.142857
= 0.571428‾0.\overline{571428}0.571428
(iv) 57\dfrac{5}{7}75 = 5×175 \times \dfrac{1}{7}5×71
= 5×0.142857‾5 \times 0.\overline{142857}5×0.142857
= 0.714285‾0.\overline{714285}0.714285
(v)67\dfrac{6}{7}76 = 6×176 \times \dfrac{1}{7}6×71
= 6×0.142857‾6 \times 0.\overline{142857}6×0.142857
= 0.857142‾0.\overline{857142}0.857142
