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How many times will the digit 7 be written when listing the
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12 Jul 2010, 20:12
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What is the best approach to solve problems such as these two examples below. If someone can solve them and show me the steps they took, I would appreciate it greatly. The GMAT Club test explanations were a little difficult for me to digest. "How many times will the digit 7 be written when listing the integers from 1 to 1000?" "How many integers between 324,700 and 458,600 have a 2 in the tens digit and a 1 in the units digit?" Answer: 1339 Thanks Club!




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Re: Digits Problem Difficulty in GMAT Club test 1
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12 Jul 2010, 20:59
tonebeeze wrote: What is the best approach to solve problems such as these two examples below. If someone can solve them and show me the steps they took, I would appreciate it greatly. The GMAT Club test explanations were a little difficult for me to digest. "How many times will the digit 7 be written when listing the integers from 1 to 1000?" "How many integers between 324,700 and 458,600 have a 2 in the tens digit and a 1 in the units digit?" Answer: 1339 Thanks Club! 1. How many times will the digit 7 be written when listing the integers from 1 to 1000?Many approaches are possible. For example: Consider numbers from 0 to 999 written as follows: 1. 000 2. 001 3. 002 4. 003 ... ... ... 1000. 999 We have 1000 numbers. We used 3 digits per number, hence used total of 3*1000=3000 digits. Now, why should ANY digit have preferences over another? We used each of 10 digits equal # of times, thus we used each digit (including 7) 3000/10=300 times. Answer: 300. 2. How many integers between 324,700 and 458,600 have a 2 in the tens digit and a 1 in the units digit?There is one number in hundred with 2 in th tens digit and 1 in the units digit: 21, 121, 221, 321, ... The difference between 324,700 and 458,600 is 458,600324,700=133,900  one number per each hundred gives 133,900/100=1,339 numbers. Answer: 1,339. Hope it's clear.
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Re: Digits Problem Difficulty in GMAT Club test 1
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14 Jul 2010, 20:57
When we write the numbers 1 to 1000 do we use 3 digits for each number?
What about: 0 1 2 3 4 5 6 7 8 9
and
10...99
The reason it works out is because we don't use zero later
so you could think of 1 as 001 and 10 as 010
However, if the question changed to how many 7's from 100 to 1000.
Then you have 900 numbers with 3 digits = 3*900 = 2700 total digits
2700 / 10 = 270
when we actually have 280 7's from 100 to 1000. right?
I came up with the same answer for the posted question 300, by counting the 7's in different groups
1 to 9= Only one 7 (7) 10 to 99= 19 (17,27,37,47,57,67,87,97 and 70, 71, 72, 73, 74, 75, 76, 78, 79 and 77 (2 sevens)) then I knew from 100 to 1000 these 20 sevens would repeat 9 more times plus all the 7's in the hundreds digits 100 sevens from 700 to 799
1 (7) +19 (tens and ones digits from 10 to 99) +180 (tens and ones digits from 100 to 1000) +100 (hundreds digits) =300



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Re: Digits Problem Difficulty in GMAT Club test 1
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15 Jul 2010, 07:26
TallJTinChina wrote: When we write the numbers 1 to 1000 do we use 3 digits for each number?
What about: 0 1 2 3 4 5 6 7 8 9
and
10...99
The reason it works out is because we don't use zero later
so you could think of 1 as 001 and 10 as 010
However, if the question changed to how many 7's from 100 to 1000.
Then you have 900 numbers with 3 digits = 3*900 = 2700 total digits
2700 / 10 = 270
when we actually have 280 7's from 100 to 1000. right?
I came up with the same answer for the posted question 300, by counting the 7's in different groups
1 to 9= Only one 7 (7) 10 to 99= 19 (17,27,37,47,57,67,87,97 and 70, 71, 72, 73, 74, 75, 76, 78, 79 and 77 (2 sevens)) then I knew from 100 to 1000 these 20 sevens would repeat 9 more times plus all the 7's in the hundreds digits 100 sevens from 700 to 799
1 (7) +19 (tens and ones digits from 10 to 99) +180 (tens and ones digits from 100 to 1000) +100 (hundreds digits) =300 This approach worked because when we write the numbers from 0 to 999 in the form XXX each digit take the values from 0 to 9 which provides that in the end all digits are used equal # of times. For the range 100 to 999 it won't be so. We can solve for this range in the following way: XX7  7 in the units place  first digit can take 9 values (from 1 to 9) and second digit can take 10 values (from 0 to 9) > total numbers with 7 in the units place: 9*10=90; X7X  7 in the tens place  first digit can take 9 values (from 1 to 9) and third digit can take 10 values (from 0 to 9) > total numbers with 7 in the tens place: 9*10=90; 7XX  7 in the hundreds place  second digit can take 10 values (from 0 to 9) and third digit can take 10 values (from 0 to 9) > total numbers with 7 in the hundreds place: 10*10=100. TOTAL: 90+90+100=280. Hope it helps.
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New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



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Re: Digits Problem Difficulty in GMAT Club test 1
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14 Jan 2012, 21:44
Each hundred can have 1 such number with unit digit as 1 and Tenth digit as 2, like 21, 121, 321 so we need to find number of hundred between 2 numbers, Answer 1339.



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Re: Digits Problem Difficulty in GMAT Club test 1
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10 Aug 2013, 18:17
Bunuel wrote: TallJTinChina wrote: When we write the numbers 1 to 1000 do we use 3 digits for each number?
What about: 0 1 2 3 4 5 6 7 8 9
and
10...99
The reason it works out is because we don't use zero later
so you could think of 1 as 001 and 10 as 010
However, if the question changed to how many 7's from 100 to 1000.
Then you have 900 numbers with 3 digits = 3*900 = 2700 total digits
2700 / 10 = 270
when we actually have 280 7's from 100 to 1000. right?
I came up with the same answer for the posted question 300, by counting the 7's in different groups
1 to 9= Only one 7 (7) 10 to 99= 19 (17,27,37,47,57,67,87,97 and 70, 71, 72, 73, 74, 75, 76, 78, 79 and 77 (2 sevens)) then I knew from 100 to 1000 these 20 sevens would repeat 9 more times plus all the 7's in the hundreds digits 100 sevens from 700 to 799
1 (7) +19 (tens and ones digits from 10 to 99) +180 (tens and ones digits from 100 to 1000) +100 (hundreds digits) =300 This approach worked because when we write the numbers from 0 to 999 in the form XXX each digit take the values from 0 to 9 which provides that in the end all digits are used equal # of times. For the range 100 to 999 it won't be so. We can solve for this range in the following way: XX7  7 in the units place  first digit can take 9 values (from 1 to 9) and second digit can take 10 values (from 0 to 9) > total numbers with 7 in the units place: 9*10=90; X7X  7 in the tens place  first digit can take 9 values (from 1 to 9) and third digit can take 10 values (from 0 to 9) > total numbers with 7 in the tens place: 9*10=90; 7XX  7 in the hundreds place  second digit can take 10 values (from 0 to 9) and third digit can take 10 values (from 0 to 9) > total numbers with 7 in the hundreds place: 10*10=100. TOTAL: 90+90+100=280. Hope it helps. Are the numbers are not getting repeated in the above three ranges? For example , the no 777 will be part of all the three ranges above and is being counted thrice. Please correct me if I am wrong.



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Re: Digits Problem Difficulty in GMAT Club test 1
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11 Aug 2013, 14:10
Bunuel wrote: tonebeeze wrote: What is the best approach to solve problems such as these two examples below. If someone can solve them and show me the steps they took, I would appreciate it greatly. The GMAT Club test explanations were a little difficult for me to digest. "How many times will the digit 7 be written when listing the integers from 1 to 1000?" "How many integers between 324,700 and 458,600 have a 2 in the tens digit and a 1 in the units digit?" Answer: 1339 Thanks Club! 1. How many times will the digit 7 be written when listing the integers from 1 to 1000?Many approaches are possible. For example: Consider numbers from 0 to 999 written as follows: 1. 000 2. 001 3. 002 4. 003 ... ... ... 1000. 999 We have 1000 numbers. We used 3 digits per number, hence used total of 3*1000=3000 digits. Now, why should ANY digit have preferences over another? We used each of 10 digits equal # of times, thus we used each digit (including 7) 3000/10=300 times. Answer: 300. 2. How many integers between 324,700 and 458,600 have a 2 in the tens digit and a 1 in the units digit?There is one number in hundred with 2 in th tens digit and 1 in the units digit: 21, 121, 221, 321, ... The difference between 324,700 and 458,600 is 458,600324,700=133,900  one number per each hundred gives 133,900/100=1,339 numbers. Answer: 1,339. Hope it's clear. Amazing thoughts....really brilliant...........splendid.............by Bunnel. i just have one question and its about this line " Now, why should ANY digit have preferences over another?" . Do you think from 1 to 1000, 1 and 0 haven't a little priority over others in aspects of their number ? i am telling because of especially the last number that is 1000 . it's not a 3 digit number but still here....... 7 was not here that's why couldn't make problem..... By the way, one thing to tell and it's day by day i am learning lot from your uploaded files...............thanks to you again bunnel.....
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Re: How many times will the digit 7 be written when listing the
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28 Aug 2017, 03:30
keenys wrote: Bunuel wrote: TallJTinChina wrote: When we write the numbers 1 to 1000 do we use 3 digits for each number?
What about: 0 1 2 3 4 5 6 7 8 9
and
10...99
The reason it works out is because we don't use zero later
so you could think of 1 as 001 and 10 as 010
However, if the question changed to how many 7's from 100 to 1000.
Then you have 900 numbers with 3 digits = 3*900 = 2700 total digits
2700 / 10 = 270
when we actually have 280 7's from 100 to 1000. right?
I came up with the same answer for the posted question 300, by counting the 7's in different groups
1 to 9= Only one 7 (7) 10 to 99= 19 (17,27,37,47,57,67,87,97 and 70, 71, 72, 73, 74, 75, 76, 78, 79 and 77 (2 sevens)) then I knew from 100 to 1000 these 20 sevens would repeat 9 more times plus all the 7's in the hundreds digits 100 sevens from 700 to 799
1 (7) +19 (tens and ones digits from 10 to 99) +180 (tens and ones digits from 100 to 1000) +100 (hundreds digits) =300 This approach worked because when we write the numbers from 0 to 999 in the form XXX each digit take the values from 0 to 9 which provides that in the end all digits are used equal # of times. For the range 100 to 999 it won't be so. We can solve for this range in the following way: XX7  7 in the units place  first digit can take 9 values (from 1 to 9) and second digit can take 10 values (from 0 to 9) > total numbers with 7 in the units place: 9*10=90; X7X  7 in the tens place  first digit can take 9 values (from 1 to 9) and third digit can take 10 values (from 0 to 9) > total numbers with 7 in the tens place: 9*10=90; 7XX  7 in the hundreds place  second digit can take 10 values (from 0 to 9) and third digit can take 10 values (from 0 to 9) > total numbers with 7 in the hundreds place: 10*10=100. TOTAL: 90+90+100=280. Hope it helps. Are the numbers are not getting repeated in the above three ranges? For example , the no 777 will be part of all the three ranges above and is being counted thrice. Please correct me if I am wrong. I have the same confusion. Can you explain more?



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Re: How many times will the digit 7 be written when listing the
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23 Oct 2018, 21:16
Bunuel "How many integers between 324,700 and 458,600 have a 2 in the tens digit and a 1 in the units digit?" Please correct my approach  number of digits is 6 = XXXXXX for numbers from 324700 till the last number starting with 3.... combinations : XXXXX1 ... only 1 way to fill the unit digit XXXX11 .... tens digit can 2 (so only 1 way) XXX311... hundreds digit can be 7,8,9 (3 ways) XX6311... thousands digit can be 4,5,6,7,8,9 (6 ways) X86311... ten thousands digit ca be 2,...9 (8 ways) 186311... hundred thousands digit can be only 3 ( 1 way) total digits starting with 3 = 1*8*6*3*1*1 = 144 similarly for numbers starting with 4 = 1*6*9*7*1*1 = 378 total = 378+144 i guess i left some numbers out... could you please tell me if this is the right way to approach?




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