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3 appears how many times?

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3 appears how many times?

by vinay1983 » Sun Aug 25, 2013 5:30 am
How many times will the digit "3" be written while numbering the pages of a book from 1 to 1000?

I am interested in the various types of methods to solve such problems?(Apart from the solution)

OA after some discussion
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by Brent@GMATPrepNow » Sun Aug 25, 2013 5:42 am
vinay1983 wrote:How many times will the digit "3" be written while numbering the pages of a book from 1 to 1000?
There are 1000 pages to number.
Let's add some zeros in the tens and hundreds positions, so we get page numbers like 006, 014 and 088. I suggest that we do this so that students can see that there must be an even distribution of the digits 0, 1, 2 etc.

First consider the units digit.
1/10 of all of the units digits written will be a "3"
Since there are 1000 pages to number, a "3" will be written as the units digit 100 times.

Now consider the tens digit.
1/10 of all of the tens digits written will be a "3"
Since there are 1000 pages to number, a "3" will be written as the tens digit 100 times.

Now consider the hundreds digit.
Using the same logic as above, a "3" will be written as the hundreds digit 100 times.

So, in total, a "3" will be written 100+100+100 times.

Answer: 300

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by Brent@GMATPrepNow » Sun Aug 25, 2013 5:49 am
vinay1983 wrote:How many times will the digit "3" be written while numbering the pages of a book from 1 to 1000?
Here's a slight twist to my solution above:

Let's consider how many times a "3" will be written if we number the pages from 0 to 999. As you can see, the answer will still be the same. All we have done is replaced the 1000 with 0, and neither of these have a "3"

Once again, let's add some zeros in the tens and hundreds positions, so we get page numbers like 006, 014 and 088. This way, we have THREE DIGITS PER PAGE.

If there are 1000 integers from 0 to 999, and each integer has 3 digits, then there is a total of 3000 digits written on the pages.

Since 1/10 of these 3000 digits is a "3," we can conclude that a "3" will be written 300 times.

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Brent
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