Question Number 4443 by 123456 last updated on 27/Jan/16
$$\mathrm{find}\:\mathrm{all}\:{x},{y}\in\mathbb{Z}\:\mathrm{such}\:\mathrm{that} \\ $$$${x}\in\left[\mathrm{0},\mathrm{50}\right] \\ $$$${y}\in\left[\mathrm{0},\mathrm{50}\right] \\ $$$${x}+{y}={k},{k}\in\left[\mathrm{0},\mathrm{50}\right] \\ $$$$\frac{{x}}{{x}+{y}}=\frac{\mathrm{99}}{\mathrm{100}} \\ $$
Answered by RasheedSindhi last updated on 27/Jan/16
$$\:\:^{{Rasheed}\:{Soomro}} \\ $$$$\frac{{x}}{{x}+{y}}=\frac{\mathrm{99}}{\mathrm{100}}………..{Given} \\ $$$$\mathrm{100}{x}=\mathrm{99}{x}+\mathrm{99}{y} \\ $$$${x}=\mathrm{99}{y}……………….\left({i}\right) \\ $$$${x}+{y}={k},{k}\in\left[\mathrm{0},\mathrm{50}\right]……{Given} \\ $$$${x}={k}−{y}……………..\left({ii}\right) \\ $$$${From}\:\left({i}\right)\:\&\:\left({ii}\right): \\ $$$${y}=\mathrm{0}.\mathrm{01}{k}\:\:\&\:{x}=\mathrm{0}.\mathrm{99}{k} \\ $$$${k}=\mathrm{0}\Rightarrow{x}+{y}=\mathrm{0} \\ $$$${k}=\mathrm{50}\Rightarrow.\mathrm{01}×\mathrm{50}+\mathrm{0}.\mathrm{99}×\mathrm{50}=\mathrm{50} \\ $$$$\:\:\:\:\:\left({x},{y}\:\in\left\{\mathrm{0}\right\}\cup{Z}^{+} \Rightarrow\:{k}\in\left\{\mathrm{0}\right\}\cup{Z}^{+} \right) \\ $$$$\forall\:{k}\in\left[\mathrm{0},\mathrm{50}\right],\:{both}\:{conditions}\:{satisfied}. \\ $$$$\therefore\:\left({x},{y}\right)=\left(\mathrm{0}.\mathrm{99}{k},\mathrm{0}.\mathrm{01}{k}\right)\:\forall{k}\in\left\{\mathrm{0},\mathrm{1},\mathrm{2},…,\mathrm{50}\right\} \\ $$$$\left({Total}\:\mathrm{51}\:{pairs}\right) \\ $$