Question Number 75033 by naka3546 last updated on 06/Dec/19
$$\frac{\mathrm{4}}{\mathrm{11}}\:<\:\frac{{x}}{{y}}\:<\:\frac{\mathrm{3}}{\mathrm{8}} \\ $$$${x},\:{y}\:\:\in\:\:\mathbb{Z}^{+} \\ $$$${min}\:\left\{{x}+{y}\right\}\:\:=\:\:? \\ $$
Answered by mr W last updated on 06/Dec/19
$$\frac{\mathrm{4}}{\mathrm{11}}<\frac{{x}}{{y}}<\frac{\mathrm{3}}{\mathrm{8}} \\ $$$$\frac{\mathrm{8}{x}}{\mathrm{3}}<{y}<\frac{\mathrm{11}{x}}{\mathrm{4}} \\ $$$$\lfloor\frac{\mathrm{8}{x}}{\mathrm{3}}\rfloor+\mathrm{1}\leqslant{y}\leqslant\lceil\frac{\mathrm{11}{x}}{\mathrm{4}}\rceil−\mathrm{1} \\ $$$$\lfloor\frac{\mathrm{8}{x}}{\mathrm{3}}\rfloor+\mathrm{1}\leqslant\lceil\frac{\mathrm{11}{x}}{\mathrm{4}}\rceil−\mathrm{1} \\ $$$$\Rightarrow{x}_{{min}} =\mathrm{7} \\ $$$$\Rightarrow{y}_{{min}} =\mathrm{19} \\ $$$$\left({x}+{y}\right)_{{min}} =\mathrm{7}+\mathrm{19}=\mathrm{26} \\ $$$$ \\ $$$${the}\:{next}\:{ones}\:{are}: \\ $$$$\frac{\mathrm{7}}{\mathrm{19}},\:\frac{\mathrm{10}}{\mathrm{27}},\:\frac{\mathrm{11}}{\mathrm{30}},\:\frac{\mathrm{13}}{\mathrm{35}},\:\frac{\mathrm{16}}{\mathrm{43}},\:… \\ $$