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

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Manager
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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?"

Thanks Club!

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Posts: 41698

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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?"

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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Re: Digits Problem Difficulty in GMAT Club test 1 [#permalink]

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14 Jul 2010, 21:57
1
KUDOS
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

Kudos [?]: 71 [1], given: 4

Math Expert
Joined: 02 Sep 2009
Posts: 41698

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Re: Digits Problem Difficulty in GMAT Club test 1 [#permalink]

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15 Jul 2010, 08:26
2
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Expert's post
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.
_________________

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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?

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?"

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.....
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Asif vai.....

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Re: How many times will the digit 7 be written when listing the [#permalink]

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15 Aug 2014, 16:50
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Re: How many times will the digit 7 be written when listing the [#permalink]

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07 Feb 2016, 01:07
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Re: How many times will the digit 7 be written when listing the [#permalink]

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19 Jul 2017, 19:20
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: How many times will the digit 7 be written when listing the [#permalink]

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28 Aug 2017, 04:30
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?

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Re: How many times will the digit 7 be written when listing the   [#permalink] 28 Aug 2017, 04:30
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