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Question Number 4443 by 123456 last updated on 27/Jan/16
find all x,y∈Z such that  x∈[0,50]  y∈[0,50]  x+y=k,k∈[0,50]  (x/(x+y))=((99)/(100))
$$\mathrm{find}\:\mathrm{all}\:{x},{y}\in\mathbb{Z}\:\mathrm{such}\:\mathrm{that} \\ $$$${x}\in\left[\mathrm{0},\mathrm{50}\right] \\ $$$${y}\in\left[\mathrm{0},\mathrm{50}\right] \\ $$$${x}+{y}={k},{k}\in\left[\mathrm{0},\mathrm{50}\right] \\ $$$$\frac{{x}}{{x}+{y}}=\frac{\mathrm{99}}{\mathrm{100}} \\ $$
Answered by RasheedSindhi last updated on 27/Jan/16
 ^(Rasheed Soomro)   (x/(x+y))=((99)/(100))...........Given  100x=99x+99y  x=99y...................(i)  x+y=k,k∈[0,50]......Given  x=k−y.................(ii)  From (i) & (ii):  y=0.01k  & x=0.99k  k=0⇒x+y=0  k=50⇒.01×50+0.99×50=50       (x,y ∈{0}∪Z^+ ⇒ k∈{0}∪Z^+ )  ∀ k∈[0,50], both conditions satisfied.  ∴ (x,y)=(0.99k,0.01k) ∀k∈{0,1,2,...,50}  (Total 51 pairs)
$$\:\:^{{Rasheed}\:{Soomro}} \\ $$$$\frac{{x}}{{x}+{y}}=\frac{\mathrm{99}}{\mathrm{100}}………..{Given} \\ $$$$\mathrm{100}{x}=\mathrm{99}{x}+\mathrm{99}{y} \\ $$$${x}=\mathrm{99}{y}……………….\left({i}\right) \\ $$$${x}+{y}={k},{k}\in\left[\mathrm{0},\mathrm{50}\right]……{Given} \\ $$$${x}={k}−{y}……………..\left({ii}\right) \\ $$$${From}\:\left({i}\right)\:\&\:\left({ii}\right): \\ $$$${y}=\mathrm{0}.\mathrm{01}{k}\:\:\&\:{x}=\mathrm{0}.\mathrm{99}{k} \\ $$$${k}=\mathrm{0}\Rightarrow{x}+{y}=\mathrm{0} \\ $$$${k}=\mathrm{50}\Rightarrow.\mathrm{01}×\mathrm{50}+\mathrm{0}.\mathrm{99}×\mathrm{50}=\mathrm{50} \\ $$$$\:\:\:\:\:\left({x},{y}\:\in\left\{\mathrm{0}\right\}\cup{Z}^{+} \Rightarrow\:{k}\in\left\{\mathrm{0}\right\}\cup{Z}^{+} \right) \\ $$$$\forall\:{k}\in\left[\mathrm{0},\mathrm{50}\right],\:{both}\:{conditions}\:{satisfied}. \\ $$$$\therefore\:\left({x},{y}\right)=\left(\mathrm{0}.\mathrm{99}{k},\mathrm{0}.\mathrm{01}{k}\right)\:\forall{k}\in\left\{\mathrm{0},\mathrm{1},\mathrm{2},…,\mathrm{50}\right\} \\ $$$$\left({Total}\:\mathrm{51}\:{pairs}\right) \\ $$

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