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
$$\frac{\mathrm{4}}{\mathrm{11}}<\frac{{x}}{{y}}<\frac{\mathrm{3}}{\mathrm{8}} \\ $$ $$\frac{\mathrm{8}{x}}{\mathrm{3}}<{y}<\frac{\mathrm{11}{x}}{\mathrm{4}} \\ $$ $$\lfloor\frac{\mathrm{8}{x}}{\mathrm{3}}\rfloor+\mathrm{1}\leqslant{y}\leqslant\lceil\frac{\mathrm{11}{x}}{\mathrm{4}}\rceil−\mathrm{1} \\ $$ $$\lfloor\frac{\mathrm{8}{x}}{\mathrm{3}}\rfloor+\mathrm{1}\leqslant\lceil\frac{\mathrm{11}{x}}{\mathrm{4}}\rceil−\mathrm{1} \\ $$ $$\Rightarrow{x}_{{min}} =\mathrm{7} \\ $$ $$\Rightarrow{y}_{{min}} =\mathrm{19} \\ $$ $$\left({x}+{y}\right)_{{min}} =\mathrm{7}+\mathrm{19}=\mathrm{26} \\ $$ $$ \\ $$ $${the}\:{next}\:{ones}\:{are}: \\ $$ $$\frac{\mathrm{7}}{\mathrm{19}},\:\frac{\mathrm{10}}{\mathrm{27}},\:\frac{\mathrm{11}}{\mathrm{30}},\:\frac{\mathrm{13}}{\mathrm{35}},\:\frac{\mathrm{16}}{\mathrm{43}},\:... \\ $$