Question Number 220737 by Spillover last updated on 18/May/25
Answered by som(math1967) last updated on 18/May/25
$$\:\mathrm{0}.\overset{.} {\mathrm{6}}=\mathrm{0}.\mathrm{6}+\mathrm{0}.\mathrm{06}+\mathrm{0}.\mathrm{006}+… \\ $$$$=\frac{\mathrm{0}.\mathrm{6}}{\mathrm{1}−.\mathrm{1}}=\frac{\mathrm{6}}{\mathrm{9}}=\frac{\mathrm{2}}{\mathrm{3}} \\ $$
Commented by Spillover last updated on 18/May/25
$${correct} \\ $$
Answered by Rasheed.Sindhi last updated on 19/May/25
$${Algebraic}\:{approach}: \\ $$$${let}\:{x}=\mathrm{0}.\overset{.} {\mathrm{6}}\:\:\:\:\:…\left({i}\right) \\ $$$${Multiplying}\:{by}\:\mathrm{10}: \\ $$$$\:\:\:\:\mathrm{10}{x}=\mathrm{6}.\overset{.} {\mathrm{6}}\:\:\:…\left({ii}\right) \\ $$$$\left({ii}\right)−\left({i}\right): \\ $$$$\:\:\:\:\:\mathrm{9}{x}=\mathrm{6} \\ $$$$\:\:\:\:\:\:{x}=\frac{\mathrm{6}}{\mathrm{9}}=\frac{\mathrm{2}}{\mathrm{3}} \\ $$