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

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How many times will the digit 7 be written when listing the [#permalink] New post 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 [#permalink] New post 12 Jul 2010, 20:59
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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 458,600-324,700=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] New post 14 Jul 2010, 20:57
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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] New post 15 Jul 2010, 07:26
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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.
_________________

NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

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] New post 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 [#permalink] New post 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 [#permalink] New post 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 458,600-324,700=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 [#permalink] New post 15 Aug 2014, 15:50
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Re: How many times will the digit 7 be written when listing the   [#permalink] 15 Aug 2014, 15:50
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