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Question Number 111155 by Aina Samuel Temidayo last updated on 02/Sep/20

$$\mathrm{Find}\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{rational}\mathrm{numbers}$$ $$\mathrm{r},0<\mathrm{r}<1,\mathrm{such}\mathrm{that}\mathrm{when}\mathrm{r}\mathrm{is}\mathrm{written}$$ $$\mathrm{as}\mathrm{a}\mathrm{fraction}\mathrm{in}\mathrm{lowest}\mathrm{term}.\mathrm{The}$$ $$\mathrm{numerator}\mathrm{and}\mathrm{the}\mathrm{denominator}$$ $$\mathrm{have}\mathrm{a}\mathrm{sum}\mathrm{of}1000.$$

Answered by Her_Majesty last updated on 02/Sep/20

$$r=\frac{n}{1000-n};0<r<1\Rightarrow 0<n<500\wedge n\ne 2k\wedge n\ne 5k$$ $$between0and500:249numbersn=2kand$$ $$99numbersn=5kbutwehave49numbers$$ $$n=2\times 5k\Rightarrow 50numbersleft$$ $$sumofnotallowednumbers=299$$ $$\Rightarrow allowednumbers=200=answer$$ $$$$

Commented byAina Samuel Temidayo last updated on 02/Sep/20

$$\mathrm{Thanks}$$

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