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How many times will the digit 7 be written when listing the [#permalink]
12 Jul 2010, 20:12
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100% (00:00) correct
0% (00:00) wrong based on 2 sessions
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?"
Re: Digits Problem Difficulty in GMAT Club test 1 [#permalink]
12 Jul 2010, 20:59
7
This post received KUDOS
Expert's post
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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.
Re: Digits Problem Difficulty in GMAT Club test 1 [#permalink]
14 Jul 2010, 20:57
1
This post received KUDOS
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
Re: Digits Problem Difficulty in GMAT Club test 1 [#permalink]
15 Jul 2010, 07:26
1
This post received KUDOS
Expert's post
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.
Re: Digits Problem Difficulty in GMAT Club test 1 [#permalink]
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.
Re: Digits Problem Difficulty in GMAT Club test 1 [#permalink]
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.
Re: Digits Problem Difficulty in GMAT Club test 1 [#permalink]
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.
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..... _________________
Re: How many times will the digit 7 be written when listing the [#permalink]
15 Aug 2014, 15:50
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Re: How many times will the digit 7 be written when listing the [#permalink]
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