faulty odometer

For students of class 6-8 (age 12 to 14)
barnik
Posts:13
Joined:Wed Dec 03, 2014 3:37 pm
faulty odometer

Unread post by barnik » Fri Dec 05, 2014 3:02 pm

A car has a defected odometer(distance measuring device),it goes directly from 3 to 5,that it does not have the digit 4.As for example, when the odometer shows 39 and then travel one more kilometer,it should show 40 but instead , it shows 50. Now on a certain case the reading in odometer was 2005,determine the exact distance traveled by the car.

source- faridpur divisional math olympiad, 12th bdmo

tanmoy
Posts:312
Joined:Fri Oct 18, 2013 11:56 pm
Location:Rangpur,Bangladesh

Re: faulty odometer

Unread post by tanmoy » Mon Dec 08, 2014 5:56 pm

The answer is $1462$.
$\because$ The odometer doesn't have the digit $4$,count the number of the numbers which have at least one $4$ in it from $1$ to $2005$.There are $543$ numbers from $1$ to $2005$ which have at least one $4$ in it the car skipped them.So,the exact distance traveled by the car was $2005-543=1462$ :)
"Questions we can't answer are far better than answers we can't question"

tanmoy
Posts:312
Joined:Fri Oct 18, 2013 11:56 pm
Location:Rangpur,Bangladesh

Re: faulty odometer

Unread post by tanmoy » Mon Dec 08, 2014 6:04 pm

I have got another approach which is easier :D
Assume that the number $2005$ is in base $9$.Now,convert it to decimal.
$2005(_{9})=2\times 9^{3}+0\times 9^{2}+0\times 9^{1}+5\times 9^{0}=1463$ .But,the number $2004$ is not counted.So,the exact distance traveled by the car is $1462$ :)
"Questions we can't answer are far better than answers we can't question"

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