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Question Number 55482 by pooja24 last updated on 25/Feb/19
A group of n students are numbered  continously from first sdtudent as 1,2,3,......  If 1101 digits had to be used in all,what is the   number of students in the group
$${A}\:{group}\:{of}\:{n}\:{students}\:{are}\:{numbered} \\ $$$${continously}\:{from}\:{first}\:{sdtudent}\:{as}\:\mathrm{1},\mathrm{2},\mathrm{3},…… \\ $$$${If}\:\mathrm{1101}\:{digits}\:{had}\:{to}\:{be}\:{used}\:{in}\:{all},{what}\:{is}\:{the}\: \\ $$$${number}\:{of}\:{students}\:{in}\:{the}\:{group} \\ $$
Answered by mr W last updated on 25/Feb/19
1,2,3...,9⇒Σ=9×1=9 digits  10,11,12,...,99⇒Σ=90×2=180 digits  100,101,102,...,999⇒Σ=900×3=2700 digits  since totally 1101 digits are used,  n must be a 3−digit number  9+180+(n−99)×3=1101  ⇒n=403  i.e. there are 403 students in the group.
$$\mathrm{1},\mathrm{2},\mathrm{3}…,\mathrm{9}\Rightarrow\Sigma=\mathrm{9}×\mathrm{1}=\mathrm{9}\:{digits} \\ $$$$\mathrm{10},\mathrm{11},\mathrm{12},…,\mathrm{99}\Rightarrow\Sigma=\mathrm{90}×\mathrm{2}=\mathrm{180}\:{digits} \\ $$$$\mathrm{100},\mathrm{101},\mathrm{102},…,\mathrm{999}\Rightarrow\Sigma=\mathrm{900}×\mathrm{3}=\mathrm{2700}\:{digits} \\ $$$${since}\:{totally}\:\mathrm{1101}\:{digits}\:{are}\:{used}, \\ $$$${n}\:{must}\:{be}\:{a}\:\mathrm{3}−{digit}\:{number} \\ $$$$\mathrm{9}+\mathrm{180}+\left({n}−\mathrm{99}\right)×\mathrm{3}=\mathrm{1101} \\ $$$$\Rightarrow{n}=\mathrm{403} \\ $$$${i}.{e}.\:{there}\:{are}\:\mathrm{403}\:{students}\:{in}\:{the}\:{group}. \\ $$
Commented by tanmay.chaudhury50@gmail.com last updated on 25/Feb/19
excellent sir...
$${excellent}\:{sir}… \\ $$
Commented by mr W last updated on 25/Feb/19
thanks sir!  if m=f(n) is the total number of digits  from 1 to n. is it possible to find a  formula for f(n) in terms of n?  furthermore find n=f^(−1) (m) if   possible.
$${thanks}\:{sir}! \\ $$$${if}\:{m}={f}\left({n}\right)\:{is}\:{the}\:{total}\:{number}\:{of}\:{digits} \\ $$$${from}\:\mathrm{1}\:{to}\:{n}.\:{is}\:{it}\:{possible}\:{to}\:{find}\:{a} \\ $$$${formula}\:{for}\:{f}\left({n}\right)\:{in}\:{terms}\:{of}\:{n}? \\ $$$${furthermore}\:{find}\:{n}={f}^{−\mathrm{1}} \left({m}\right)\:{if}\: \\ $$$${possible}. \\ $$

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