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How many times will the digit 7 be written when listing the [#permalink]
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12 Jul 2010, 21:12
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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 [#permalink]
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12 Jul 2010, 21:59
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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 [#permalink]
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14 Jul 2010, 21:57
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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 [#permalink]
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15 Jul 2010, 08:26
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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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.
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Re: Digits Problem Difficulty in GMAT Club test 1 [#permalink]
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14 Jan 2012, 22:44
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 [#permalink]
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10 Aug 2013, 19:17
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 [#permalink]
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11 Aug 2013, 15:10
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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