Question 1176475: Given that a total of 149 1's are used for all the page numbers in a book, at least how many pages are there in the book?
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
There is no magic formula for working a problem like this. You just need to count the 1's that are used in numbering the pages of a book.
I will solve a similar problem using different numbers; then you can use the process to solve your problem.
Find the number of pages in a book if a total of 172 3's are used.
1-digit numbers: the digit 3 is used 1 time.
2-digit numbers: the digit 3 is used 20 times -- 10 times as a tens digit and 10 times as a ones digit
To this point we have used a total of 21 3's to number the pages.
3-digit numbers 100-199: no 3's in the hundreds column; so again 20 3's.
3-digit numbers 200-299: same as 100-199 -- 20 more 3's.
To this point we have used 61 3's to number the pages; we need to use 111 more.
3-digit numbers 300-399: To write all those page numbers, we would have the 20 3's in the tens and ones columns again, plus 100 3's in the hundreds column. That's 120 more 3's; but we only have 111 left.
So now we do some good old guessing and checking. The page numbers 390-399 use 11 3's -- 10 in the hundreds column and 1 in the ones column. 11 off of the 120 it takes to write all the page numbers to 399 leaves us with a total of 109; add that to the 61 3's we used to write the page numbers 1-299 and we see that we have used 170 3's to write the page numbers from 1 to 389.
So we need to use 2 more 3's; we will do that in writing the page numbers 390 and 391.
So our answer is that the number of pages in the book is 391 if we use 172 3's to write all the page numbers.
Now go through a similar process to solve your problem....
Answer by ikleyn(52800)  (Show Source):
You can put this solution on YOUR website! .
Given that a total of 149 1's are used for all the page numbers in a book,
at least how many pages are there in the book?
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Actually, this problem requires very accurate counting of digits - more accurate than in the solution by other tutor.
So I place this accurate solution below.
Find several distinctions from the solution of the other tutor.
(1) Consider one-digit numbers from 1 to 9 inclusive.
There is only one digit "1" in this set of numbers.
(2) Consider two-digit numbers from 10 to 99 inclusive.
In this set, there are 10 (ten) digits "1" in the "tens" position (from 10 to 19),
and 9 (nine) digits "1" in the "ones" position (11, 21, 31, . . . 91).
Thus we have 10+9 = 19 digits of "1" in two-digit numbers.
Hence, so far we have 1 + 19 = 20 digits of "1" in one-digit and two-digit numbers.
We need to get 149-20 = 129 additional digits of "1".
(3) Consider three-digit numbers from 100 to 199 inclusive (100, 101, 102, . . . , 199)
In this set, there are 100 (one hundred) digits "1" in the "hundreds" position (from 100 to 199),
10 (ten) digits "1" in the "tens" position (110, 111, 112, . . . , 119),
and 10 (ten) digits "1" in the "ones" position (101, 111, 121, . . . , 191).
Thus we have additional 100+10+10 = 120 digits "1" in three-digit numbers from 100 to 199 inclusive.
Hence, so far (arriving to 199) we have accumulated 20 + 120 = 140 digits of "1".
140 is still smaller comparing with 149, so we will continue moving forward from the number of 200.
We still should collect 9 more digits of "1".
(4) So, at the position of 199, where we have 140 digits of "1", we will move farther in slow pace, accurately counting digits "1".
The numbers from 200 to 209 inclusive add 1 digits of "1" to 140, giving the total of 140 + 1 = 141.
We still need accumulate 149-141 = 8 digits of "1"
The numbers 210, 211, 212, 213, 214, 215, 216 will add these 8 digits of "1", giving finally the total of 149 at N = 216.
ANSWER. The book has 216 pages.
CHECK. The numbers from 1 to 9 inclusive give one digit "1".
The numbers from 10 to 99 add 19 digits "1", making the cumulative sum equal to 1 + 19 = 20.
The numbers from 100 to 199 add 120 digits "1", making the cumulative sum equal to 20 + 120 = 140.
The numbers from 200 to 209 add 1 digit "1", making the cumulative sum equal to 140 + 1 = 141.
The numbers 210, 211, 212, 213, 214, 215, 216 add 8 digits of "1", making the cumulative sum equal to 141 + 8 = 149.
ANSWER. There are 216 pages in the book.
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The problem is solved and the solution is completed.
As you see, there is no any advanced super-idea in the solution.
It requires accurate counting, only.
So, let this solution serves as a TEMPLATE for you, showing how to solve problems accurately.
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