How-many-times-does-the-digit-6-appear-when-writing-from-6-to-400-

Question Number 44038 by Tawa1 last updated on 20/Sep/18

$$\mathrm{How}\:\mathrm{many}\:\mathrm{times}\:\mathrm{does}\:\mathrm{the}\:\mathrm{digit}\:\mathrm{6}\:\mathrm{appear}\:\mathrm{when}\:\mathrm{writing}\:\mathrm{from}\:\:\mathrm{6}\:\mathrm{to}\:\mathrm{400}\:? \\ $$

Answered by $@ty@m last updated on 20/Sep/18

6, 16,26,36,46,56, 60,61,62,63,64,65,66,67,68,69, 76,86,96 i.e. 20 times in between 1 to 100 ⇒80 times in between 1 (or 6) to 400

$$\mathrm{6},\:\mathrm{16},\mathrm{26},\mathrm{36},\mathrm{46},\mathrm{56}, \\ $$$$\mathrm{60},\mathrm{61},\mathrm{62},\mathrm{63},\mathrm{64},\mathrm{65},\mathrm{66},\mathrm{67},\mathrm{68},\mathrm{69}, \\ $$$$\mathrm{76},\mathrm{86},\mathrm{96} \\ $$$${i}.{e}.\:\mathrm{20}\:{times}\:{in}\:{between}\:\mathrm{1}\:{to}\:\mathrm{100} \\ $$$$\Rightarrow\mathrm{80}\:{times}\:{in}\:{between}\:\mathrm{1}\:\left({or}\:\mathrm{6}\right)\:{to}\:\mathrm{400} \\ $$

Commented by Tawa1 last updated on 20/Sep/18

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$

Commented by $@ty@m last updated on 20/Sep/18

Another way: In between 1 to 100, No. of numbers in which unit digit is 6=10 No. of numbers in which tens digit is 6=10 ∴ 6 appears 20 times in between 1 to 100.

$${Another}\:{way}: \\ $$$${In}\:{between}\:\mathrm{1}\:{to}\:\mathrm{100}, \\ $$$${No}.\:{of}\:{numbers}\:{in}\:{which}\:{unit} \\ $$$${digit}\:{is}\:\mathrm{6}=\mathrm{10} \\ $$$${No}.\:{of}\:{numbers}\:{in}\:{which}\:{tens} \\ $$$${digit}\:{is}\:\mathrm{6}=\mathrm{10} \\ $$$$\therefore\:\mathrm{6}\:{appears}\:\mathrm{20}\:{times}\:{in}\:{between}\:\mathrm{1}\:{to}\:\mathrm{100}. \\ $$

Commented by Tawa1 last updated on 20/Sep/18

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$

Answered by MrW3 last updated on 20/Sep/18

in numbers with one digit: 1 time in numbers with two digits: 9+8+1×2=19 times in numbers with three digits: 3×(2×9+1×2)=60 times Σ=1+19+60=80 times

$${in}\:{numbers}\:{with}\:{one}\:{digit}: \\ $$$$\mathrm{1}\:{time} \\ $$$$ \\ $$$${in}\:{numbers}\:{with}\:{two}\:{digits}: \\ $$$$\mathrm{9}+\mathrm{8}+\mathrm{1}×\mathrm{2}=\mathrm{19}\:{times} \\ $$$$ \\ $$$${in}\:{numbers}\:{with}\:{three}\:{digits}: \\ $$$$\mathrm{3}×\left(\mathrm{2}×\mathrm{9}+\mathrm{1}×\mathrm{2}\right)=\mathrm{60}\:{times} \\ $$$$ \\ $$$$\Sigma=\mathrm{1}+\mathrm{19}+\mathrm{60}=\mathrm{80}\:{times} \\ $$

Commented by malwaan last updated on 20/Sep/18