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# How many times will the digit 7 be written when listing the

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Manager
Status: Current MBA Student
Joined: 19 Nov 2009
Posts: 98
Concentration: Finance, General Management
GMAT 1: 720 Q49 V40
How many times will the digit 7 be written when listing the  [#permalink]

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12 Jul 2010, 20:12
1
6
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Question Stats:

50% (00:00) correct 50% (03:41) wrong based on 7 sessions

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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?"

Thanks Club!
Math Expert
Joined: 02 Sep 2009
Posts: 53066
Re: Digits Problem Difficulty in GMAT Club test 1  [#permalink]

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12 Jul 2010, 20:59
11
5
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?"

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.

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 458,600-324,700=133,900 - one number per each hundred gives 133,900/100=1,339 numbers.

Hope it's clear.
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Manager
Status: Waiting to hear from University of Texas at Austin
Joined: 24 May 2010
Posts: 67
Location: Changchun, China
Schools: University of Texas at Austin, Michigan State
Re: Digits Problem Difficulty in GMAT Club test 1  [#permalink]

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14 Jul 2010, 20:57
1
When we write the numbers 1 to 1000 do we use 3 digits for each number?

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
Math Expert
Joined: 02 Sep 2009
Posts: 53066
Re: Digits Problem Difficulty in GMAT Club test 1  [#permalink]

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15 Jul 2010, 07:26
2
TallJTinChina wrote:
When we write the numbers 1 to 1000 do we use 3 digits for each number?

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.
_________________
Intern
Joined: 04 Aug 2011
Posts: 40
Location: United States
GMAT 1: 570 Q45 V25
GPA: 4
WE: Information Technology (Computer Software)
Re: Digits Problem Difficulty in GMAT Club test 1  [#permalink]

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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.
Intern
Joined: 12 Mar 2013
Posts: 13
Re: Digits Problem Difficulty in GMAT Club test 1  [#permalink]

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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?

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.
Senior Manager
Joined: 10 Jul 2013
Posts: 313
Re: Digits Problem Difficulty in GMAT Club test 1  [#permalink]

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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?"

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.

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 458,600-324,700=133,900 - one number per each hundred gives 133,900/100=1,339 numbers.

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.....
_________________

Asif vai.....

Intern
Joined: 26 Jul 2014
Posts: 12
Re: How many times will the digit 7 be written when listing the  [#permalink]

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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?

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?
Director
Joined: 20 Sep 2016
Posts: 595
Re: How many times will the digit 7 be written when listing the  [#permalink]

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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?"

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?
Re: How many times will the digit 7 be written when listing the   [#permalink] 23 Oct 2018, 21:16
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