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How many times will the digit 7 be written when listing the [#permalink]
 21 Aug 2008, 13:07
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How many times will the digit 7 be written when listing the integers from 1 to 1000? 110 111 271 300 304 This problem is from GMACLUB test. I agree with answer but not the explanation. Let see how others will tackle this problem.
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Last edited by x2suresh on 21 Aug 2008, 14:01, edited 1 time in total.
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Re: GMAT CLUB - PS. (Dont agree with OE -but agree with OA) [#permalink]
21 Aug 2008, 13:16
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x2suresh wrote: How many times will the digit 7 be written when listing the integers from 1 to 1000?
110
111
271
300
304
This problem is from GMACLUB test. I agree with answer but not the explanation.
Let see how others will tackle this problem. D I would do it by digits place. Ones digit: For every 100 numbers, 7 appears 10 times (x07, x17, x27, x37, x47, x57, x67, x77, x87, x97) 10*10 = 100 Tens digit: For every 100 numbers, 7 appears 10 times (x70, x71, x72, x73, x74, x75, x76, x77, x78, x79) 10*10 = 100 Hundreds digit: 7 appears in the hundreds digit in every number from 700-799 799-700+1 = 100 Total 7s: 100+100+100 = 300
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Re: GMAT CLUB - PS. (Dont agree with OE -but agree with OA) [#permalink]
21 Aug 2008, 13:18
I figured it like this: (and I realize there is a formula for this too, but I'm not sure what it is). This method took me about 90 seconds to complete. Answer: 300 Units Digit: 10-7's per 100 numbers * 10 groups of 100 from 1 - 1000 = 100 Tens Digit: 10-7's in the 70's section of each group of 100 so 10*10 = 100 Hundreds digit: 100-7's from 700 to 799, so 100 again Total = 300. Am I forgetting anything? x2suresh wrote: How many times will the digit 7 be written when listing the integers from 1 to 1000?
110
111
271
300
304
This problem is from GMACLUB test. I agree with answer but not the explanation.
Let see how others will tackle this problem.
_________________
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J Allen Morris
**I'm pretty sure I'm right, but then again, I'm just a guy with his head up his a$$.
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Re: GMAT CLUB - PS. (Dont agree with OE -but agree with OA) [#permalink]
21 Aug 2008, 13:42
I got D with similar counting method and took about 90 sec
But I would love to know some shorter method
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Re: GMAT CLUB - PS. (Dont agree with OE -but agree with OA) [#permalink]
21 Aug 2008, 13:52
When counting the times a certain number appears, this might be a more scientific method of counting. If you're counting units digits, you know that any single number will appear in the units digit column 1 time out of 10. If you're counting 1 to n, then multiply the number of times that digit appears out of 10 * n/10: Example: looking for # of 5s in the units column from 1 to 600. There will be a 5 in each group of 10. So take 600 (the 1 to 600) and divide 600 by 10, for 60. There will be 60-5's in the units column from 1 to 600. Tens column: You see there will be 10-5's in the tens column per group of 100. Divide the last number by the size of the group. 600 / 100 = 6 groups, so 60-50s in the tens digit. Hundreds column: You know there will be 5's here only in 500-599. So that's 100 numbers per group of 1000. Since we don't go all the way to 1000, we know that's 1. so It's 100*1. what's the total? 60+60+100 = 220. LOL....I'm not sure this makes it any easier!! hibloom wrote: I got D with similar counting method and took about 90 sec
But I would love to know some shorter method
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Re: GMAT CLUB - PS [#permalink]
21 Aug 2008, 14:09
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Short method. Consider slightly modified set: 000 - 999 We have 3000 digits. Frequency for 7 is 1/10. Therefore, answer is 300
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Re: GMAT CLUB - PS. (Dont agree with OE -but agree with OA) [#permalink]
21 Aug 2008, 14:10
x2suresh wrote: zoinnk wrote: x2suresh wrote: How many times will the digit 7 be written when listing the integers from 1 to 1000?
110
111
271
300
304
This problem is from GMACLUB test. I agree with answer but not the explanation.
Let see how others will tackle this problem. For every 100 numbers, 7 appears 10 times (x07, x17, x27, x37, x47, x57, x67, x77, x87, x97) --> Do you belieive 7 appeared 10 times or 11 times here. Don't worry your answer is correct.. Here you treated x77 (7 in the 10th place ignored.. and reconsider when "Ten digit calculations" ) 10*10 = 100 Tens digit: For every 100 numbers, 7 appears 10 times (x70, x71, x72, x73, x74, x75, x76, x77, x78, x79) 10*10 = 100 Hundreds digit: 7 appears in the hundreds digit in every number from 700-799 799-700+1 = 100 Total 7s: 100+100+100 = 300 I ignored the 7 in the tens place for that part of the calculation because i was just counting the 7s in the ones digit.
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Re: GMAT CLUB - PS [#permalink]
21 Aug 2008, 14:10
How would you do it counting zeros between 1 and 1000? walker wrote: Short method. Consider slightly modified set: 000 - 999
We have 3000 digits. Frequency for 7 is 1/10. Therefore, answer is 300
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Re: GMAT CLUB - PS. (Dont agree with OE -but agree with OA) [#permalink]
21 Aug 2008, 14:14
Hi Allen and Zoink, Agree with both of you.. Now I agree with OE approach too.. OE is also explained similar to your approach. ( You ignored the x77 tenth digit 7 when counting unit digit calculations and reconsider this 7 when tenth digit calucations.. see zoink reply) My approach was: 7 occur only once. (7XX, X7X,XX7) = "7 is one of the digit" * "select other than 7" * "select other than 7" * (no of ways 7 can appear ) = 1*9*9* 3 =243 7 occur twice (77X,7X7,X77) = "7 is two of the digit" * "select 3rd one other than 7" * ( Each number 7 written twice) = 1*9*3 * 2 = 54 7 occur thrice = 1*3 (7 appeared 3 times) =3 Sum = 243+54+3= 300.
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Last edited by x2suresh on 21 Aug 2008, 14:15, edited 1 time in total.
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Re: GMAT CLUB - PS [#permalink]
21 Aug 2008, 14:15
I considered, for example, 007 instead of 7 in order to have the same frequency for all digits
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Re: GMAT CLUB - PS [#permalink]
21 Aug 2008, 14:25
the way i did this is as follows..
we are really looking at 3 digit number..
X.Y.Z where x or y or z can be 7 or all of them could be 7..
howerver indepndently there are only 10 possibilities for x, 10 for y and 10 for z..
therefore there is only 1 possibility for x=7 and 1 possibility for y=7 and 1 possibility for z=7.
10.10.1 if z=7
10.1.10 if y=7
1.10.10 if x=7
total number of possibilities=100+100+100=300
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Re: GMAT CLUB - PS [#permalink]
21 Aug 2008, 15:32
Single digit nos.- 1 seven
Double digit nos.- 1*10 + 9*1 =19 sevens
3-digit nos.- 1*10*10 + 9*1*10 + 9*10*1 = 280 sevens
Total = 1 + 19 + 280 = 300 sevens
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Re: GMAT CLUB - PS [#permalink]
23 Aug 2008, 14:25
walker wrote: Short method. Consider slightly modified set: 000 - 999
We have 3000 digits. Frequency for 7 is 1/10. Therefore, answer is 300 I think this is genius Man you are really good
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Re: GMAT CLUB - PS
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23 Aug 2008, 14:25
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