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Question Number 56301 by naka3546 last updated on 13/Mar/19

$$Minimumvalueofbthatsatisfythe$$$$followinginequality$$$$\frac{53}{201}<\frac{a}{b}<\frac{4}{15}is...$$

Commented by naka3546 last updated on 13/Mar/19

$$a,barepositiveintegers.$$

Answered by tanmay.chaudhury50@gmail.com last updated on 13/Mar/19

$$201=3\times 6715=3\times 5$$$$\frac{53}{201}=\frac{265}{1005}and\frac{4}{15}=\frac{4\times 67}{15\times 67}=\frac{268}{1005}$$$$\frac{a}{b}=\frac{266}{1005}\leftarrow firstanswer$$$$\frac{a}{b}=\frac{267}{1005}\leftarrow secondanswer$$$$\mathit{s}\mathit{o}\mathit{b}=1005$$

Commented by mr W last updated on 14/Mar/19

$$withmininmumb:\frac{9}{34}$$$$thenextare\frac{13}{49},\frac{14}{53},\frac{17}{64},\frac{19}{72}....$$

Commented by naka3546 last updated on 14/Mar/19

$$howtogetit,sir.$$

Commented by MJS last updated on 14/Mar/19

$$\frac{53}{201}<\frac{a}{b}<\frac{4}{15}$$$$\frac{53}{201}b<a<\frac{4}{15}b$$$$b\ne 15n\wedge b\ne 201n$$$$\lceil \frac{53}{201}b\rceil >\lfloor \frac{4}{15}b\rfloor \mathrm{xor}\lceil \frac{53}{201}b\rceil =\lfloor \frac{4}{15}b\rfloor =a$$$$\mathrm{now}\mathrm{we}\mathrm{must}\mathrm{try}$$

Commented by Kunal12588 last updated on 13/Mar/19

$$sir\frac{53}{201}<\frac{33}{125}<\frac{4}{15}$$$${don}^{\prime }tknowtherecanbesmalleranswers.$$

Commented by naka3546 last updated on 14/Mar/19

$$\frac{9}{34}maybe,I{don}^{\prime }tknowaboutit.$$

Answered by mr W last updated on 15/Mar/19

$$thisismymethod:$$$$\frac{53}{201}<\frac{a}{b}<\frac{4}{15}$$$$\Rightarrow \frac{201}{53}>\frac{b}{a}>\frac{15}{4}$$$$\Rightarrow \frac{15a}{4}<b<\frac{201a}{53}$$$$\Rightarrow \lfloor \frac{15a}{4}\rfloor +1⩽b⩽\lceil \frac{201a}{53}\rceil -1$$$$$$$$foragivenvalueofawecalculate$$$$\lfloor \frac{15a}{4}\rfloor +1and\lceil \frac{201a}{53}\rceil -1.$$$$if\lfloor \frac{15a}{4}\rfloor +1⩽\lceil \frac{201a}{53}\rceil -1,thenbexists,$$$$andb=\lfloor \frac{15a}{4}\rfloor +1...\lceil \frac{201a}{53}\rceil -1.$$$$$$$$ifwestartwitha=1,2,3,...{we}^{\prime }llgetall$$$$possible\left(infinite\right)solutions:$$$$a=9\Rightarrow b=34$$$$a=13\Rightarrow b=49$$$$a=14\Rightarrow b=53$$$$a=17\Rightarrow b=64$$$$a=19\Rightarrow b=72$$$$a=21\Rightarrow b=79$$$$a=22\Rightarrow b=83$$$$a=23\Rightarrow b=87$$$$a=24\Rightarrow b=91$$$$a=25\Rightarrow b=94$$$$a=29\Rightarrow b=109$$$$a=30\Rightarrow b=113$$$$a=31\Rightarrow b=117$$$$a=32\Rightarrow b=121$$$$a=33\Rightarrow b=124,125$$$$a=35\Rightarrow b=132$$$$......$$$$a=117\Rightarrow b=439,440,442,443$$$$......$$

Commented by mr W last updated on 15/Mar/19

$$isthereaneasierwaythanthis?$$

Commented by MJS last updated on 16/Mar/19

$$\mathrm{no}.$$

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