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Intern  Joined: 26 Feb 2011
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How many integers between 324,700 and 458,600 have tens  [#permalink]

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Question Stats: 73% (01:46) correct 27% (02:19) wrong based on 1057 sessions

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How many integers between 324,700 and 458,600 have tens digit 1 and units digit 3?

A. 10,300
B. 10,030
C. 1,353
D. 1,352
E. 1,339

Originally posted by Madelaine88 on 11 Mar 2011, 08:47.
Last edited by Bunuel on 10 Aug 2012, 14:06, edited 1 time in total.
Edited the question.
Math Expert V
Joined: 02 Sep 2009
Posts: 64891
How many integers between 324,700 and 458,600 have tens  [#permalink]

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11
34
How many integers between 324,700 and 458,600 have tens digit 1 and units digit 3?
A/ 10,300
B/ 10,030
C/ 1,353
D/ 1,352
E/ 1,339

There is one number in hundred with 1 in the tens digit and 3 in the units digit: 13, 113, 213, 313, ...

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.

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How many integers between 324,700 and 458,600 have tens  [#permalink]

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Top Contributor
How many integers between 324,700 and 458,600 have tens digit 1 and units digit 3?

A. 10,300
B. 10,030
C. 1,353
D. 1,352
E. 1,339

A nice rule says: the number of integers from x to y inclusive equals y - x + 1
So, the number of integers from 324,700 to 458,600 = 458,600 - 324,700 + 1 = 133901
ASIDE: We can safely round this number 133900 (you'll see why shortly)

Now let's look at how frequently we have a 1 in the tens digit and 3 in the units digit.
Integers from 1 to 100: 13 (1 integer)
Integers from 101 to 200: 113 (1 integer)
Integers from 201 to 300: 213 (1 integer)
Integers from 301 to 400: 313 (1 integer)
.
.
.
As we can see, for every 100 integers, there's 1 integer with a 1 in the tens digit and 3 in the units digit.
In other words, 1/100 of all integers have a 1 in the tens digit and 3 in the units digit.

So, among the 133900 integers in question 1/100 of them meet the given condition.
(1/100)(133900) = 1339

Cheers,
Brent
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Originally posted by BrentGMATPrepNow on 27 Feb 2018, 12:02.
Last edited by BrentGMATPrepNow on 01 Jul 2020, 10:23, edited 2 times in total.
Director  Joined: 22 Mar 2011
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Re: HELP Me on This . . . .  [#permalink]

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mothergoose wrote:
How many integers between 324, 700 and 458, 600 have tens digit 1 and units digit 3 ?

1) 10, 300
2) 10,030
3) 1,353
4) 1,352
5)1,339

The numbers must be of the form ABCD13, where ABCD is any four-digit number between 3247 and 4585 inclusive.
ABCD cannot be 4586, as 458,613>458,600.

Therefore, the total number is 4585 - 3247 + 1 = 1339.

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is there another way to solve this?

I once saw someone solving this kind of questions by multiplying the amount of different number of digits u can put in each place (in each different digit)
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144144 wrote:
is there another way to solve this?

I once saw someone solving this kind of questions by multiplying the amount of different number of digits u can put in each place (in each different digit)

You can but you would have to take multiple cases and that would be really cumbersome. The left most digit can be either a 3 or a 4 but then next digit depends on what the leftmost digit is. If left most digit is 3, next digit can be anything from 2 to 9. If the leftmost digit is 4, the next digit can be from 0 to 5. Similarly other digits too.
So preferably, focus on the approach given by Bunuel.
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Re: How many integers between 324,700 and 458,600 have tens  [#permalink]

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+1 E

We have to count how many hundreds are from 324,700 to 458,600.
We substract 458,600 - 324,700 = 133,900
We divide, 133,900/100 = 1,339
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Intern  Joined: 11 Feb 2013
Posts: 11
How many integers between 324,700 and 458,600 have tens digit of  [#permalink]

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briks123 wrote:
Definitely A.

458,600-324,700=133,900-1=133,899.

So there are 133,899 integers between the two numbers. Now how many hundreds of numbers are there? There are 1338+1 (to account for the number 133,821).

m clear with the statement that there are 133,899 integers between the two numbers.
but how to calculate hundreds of numbers in them.....

There are 1338+1 (to account for the number 133,821-------cud not understand this particular step....

If anyone can explain me this step in little detail then plz explain coz i dont hv any idea Thanks in advance! Verbal Forum Moderator B
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Perhaps wrote:
briks123 wrote:
Definitely A.

458,600-324,700=133,900-1=133,899.

So there are 133,899 integers between the two numbers. Now how many hundreds of numbers are there? There are 1338+1 (to account for the number 133,821).

m clear with the statement that there are 133,899 integers between the two numbers.
but how to calculate hundreds of numbers in them.....

There are 1338+1 (to account for the number 133,821-------cud not understand this particular step....

If anyone can explain me this step in little detail then plz explain coz i dont hv any idea Thanks in advance! When we have to find the numbers between A and B(where both A and B are included) = (B-A)/1 +1.

When we have to find the numbers between A and B(where both A and B are excluded) = (B-A)/1 -1.

When we have to find the numbers between A and B(where either A or B is included) = (B-A)/1 .

For a number to keep having a 2 in the tens place and 1 in the units place, we have to keep adding 100 to that number. Thus, starting from 324,721, keep adding 100 to get the same units and digits place. Now to find the hundreds, all we have to do is find the number of hundreds between 458,521 and 324,721, where both are included : (458521-324721)/100+1 = 1338+1 = 1339.
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vinaymimani wrote:
Perhaps wrote:
briks123 wrote:
Definitely A.

458,600-324,700=133,900-1=133,899.

So there are 133,899 integers between the two numbers. Now how many hundreds of numbers are there? There are 1338+1 (to account for the number 133,821).

m clear with the statement that there are 133,899 integers between the two numbers.
but how to calculate hundreds of numbers in them.....

There are 1338+1 (to account for the number 133,821-------cud not understand this particular step....

If anyone can explain me this step in little detail then plz explain coz i dont hv any idea Thanks in advance! When we have to find the numbers between A and B(where both A and B are included) = (B-A)/1 +1.

When we have to find the numbers between A and B(where both A and B are excluded) = (B-A)/1 -1.

When we have to find the numbers between A and B(where either A or B is included) = (B-A)/1 .

For a number to keep having a 2 in the tens place and 1 in the units place, we have to keep adding 100 to that number. Thus, starting from 324,721, keep adding 100 to get the same units and digits place. Now to find the hundreds, all we have to do is find the number of hundreds between 458,521 and 324,721, where both are included : (458521-324721)/100+1 = 1338+1 = 1339.

what a wonderful explaination!!! even after reading all the above solutions , things dint go in my mind ....... but ur explaination ( specially the first three lines) made my basic concepts vivid.

thanks a lot for ur efforts Vinay .....

A big thanks to u Intern  Joined: 22 Feb 2014
Posts: 24
Re: How many integers between 324,700 and 458,600 have tens  [#permalink]

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Bunuel wrote:
How many integers between 324,700 and 458,600 have tens digit 1 and units digit 3?
A/ 10,300
B/ 10,030
C/ 1,353
D/ 1,352
E/ 1,339

There is one number in hundred with 1 in th tens digit and 3 in the units digit: 13, 113, 213, 313, ...

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.

Hi
Why you divided by 100..Please explain..I read all below comments but still can not understand the concept.

Math Expert V
Joined: 02 Sep 2009
Posts: 64891
Re: How many integers between 324,700 and 458,600 have tens  [#permalink]

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GGMAT730 wrote:
Bunuel wrote:
How many integers between 324,700 and 458,600 have tens digit 1 and units digit 3?
A/ 10,300
B/ 10,030
C/ 1,353
D/ 1,352
E/ 1,339

There is one number in hundred with 1 in the tens digit and 3 in the units digit: 13, 113, 213, 313, ...

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.

Hi
Why you divided by 100..Please explain..I read all below comments but still can not understand the concept.

In each hundred there is only one number with 1 in the tens digit and 3 in the units digit. How many hundreds are in 133,900? 133,900/100=1,339. One number per hundred gives total of 1,339*1 numbers.
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Intern  B
Joined: 17 Apr 2013
Posts: 34
Re: How many integers between 324,700 and 458,600 have tens  [#permalink]

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I guess possibility of 1 in the tens is 1/10 and possibility of 23 in the unit digit is 1/10, so number of the way is 458,600-324,700=133,900/100 = 1,339

Intern  Joined: 02 Feb 2011
Posts: 35
Re: How many integers between 324,700 and 458,600 have tens  [#permalink]

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Dear Bunuel,Karishma/others

I might be overthinking here.

Getting confused. I reached till the step of calculating hundreds i.e 458600-324700 = 133900;
Next I got confused and was trying to find hunders using Last-First +1 --> 133901

How are the above two concepts different.
If we take 1-200 range to calculate 100's we can do 200-1 +1 --> total integers/100

Kindly help in clarifying.
Intern  Joined: 05 Jul 2014
Posts: 5
Re: How many integers between 324,700 and 458,600 have tens  [#permalink]

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How many integers between 324,700 and 458,600 have tens digit 1 and units digit 3?

A. 10,300
B. 10,030
C. 1,353
D. 1,352
E. 1,339

I think there is an easier way to answer this question. Please let me know if I am wrong.

As you can see there are 4 different units digits in the answer choice. So It would help to find the units digit.

To get the units digits we know between 324,700 and 458,600. There are 9 numbers that fall under the requirements asked in the question.

xxx,713
xxx,813
xxx,913
xx1,013
xx1,113
xx1,213
xx1,313
xx1,413
xx1,513

So with that fact we get E as the answer.
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Re: How many integers between 324,700 and 458,600 have tens  [#permalink]

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Hi,

I have 3 questions.

1/ Here is how I approach the problem. Please check if it is correct.
Find multiples of 100 between 324,700 and 458,600: (458,600-324,700)/100 - 1 = 1339-1=1338
There are 1338 multiples of 100 starting from 324800. However, since 324700 is a legitimate multiple of 100 (because 324721 satisfies the stem) but 458600 is not (458621>458600), we have 1338+1=1339 as the answer. Is there anything incorrect?

2/ I dont understand why Bunuel subtract 324700 from 458600. What does 133900 refer to?

3/ I found this type of question quite challenging. Can anyone post the links to similar gmat problems in which the same concept is tested?

Thank you very much!
Math Expert V
Joined: 02 Aug 2009
Posts: 8738
Re: How many integers between 324,700 and 458,600 have tens  [#permalink]

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truongynhi wrote:
Hi,

I have 3 questions.

1/ Here is how I approach the problem. Please check if it is correct.
Find multiples of 100 between 324,700 and 458,600: (458,600-324,700)/100 - 1 = 1339-1=1338
There are 1338 multiples of 100 starting from 324800. However, since 324700 is a legitimate multiple of 100 (because 324721 satisfies the stem) but 458600 is not (458621>458600), we have 1338+1=1339 as the answer. Is there anything incorrect?

2/ I dont understand why Bunuel subtract 324700 from 458600. What does 133900 refer to?

3/ I found this type of question quite challenging. Can anyone post the links to similar gmat problems in which the same concept is tested?

Thank you very much!

Hi,
you are also doing the same thing what has been done in prior solution..
we have to find numbers that have 13 in the end..
324700 will have 324713..
324800 will have 324813..

so The number of 100s from 324700 to 458600 is found by subtrcting these two and then dividing by 100..
so difference in numbers= 458600-324700=133900..
How many 100s does it contain 133900/100=1339
each of these 1339 100s has one 13 in it so ans 1339

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Re: How many integers between 324,700 and 458,600 have tens  [#permalink]

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Bunuel wrote:
How many integers between 324,700 and 458,600 have tens digit 1 and units digit 3?
A/ 10,300
B/ 10,030
C/ 1,353
D/ 1,352
E/ 1,339

There is one number in hundred with 1 in the tens digit and 3 in the units digit: 13, 113, 213, 313, ...

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.

Bunel
I understood the concept of for every hundred digits we will have only one digit with tens digit 1 and units digit 3. But The number of digits between 324,700 and 458,600 should be (458,600-324,700)-1=133,900-1=133899. You have calculated it for only 133900 how?
Intern  Joined: 20 Oct 2014
Posts: 5
How many integers between 324,700 and 458,600 have tens  [#permalink]

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Hi chetan2u,

If I want to calculate how many integers between 324,700 and 458,600 have tens digits 1 (without the unit digit restriction), how can I use the same approach?

Would it be 458,600 - 324,700 = (133,900 / 100 (happens every 100 numbers)) * 10 (happens 10 times every 100) = 133,900/10 = 13,390

Could you let me know if I am doing right, please? I want to grasp the concept for other problems (if you have other problems I can use to practice, I would appreciate it).
Thank you!

chetan2u wrote:
truongynhi wrote:
Hi,

I have 3 questions.

1/ Here is how I approach the problem. Please check if it is correct.
Find multiples of 100 between 324,700 and 458,600: (458,600-324,700)/100 - 1 = 1339-1=1338
There are 1338 multiples of 100 starting from 324800. However, since 324700 is a legitimate multiple of 100 (because 324721 satisfies the stem) but 458600 is not (458621>458600), we have 1338+1=1339 as the answer. Is there anything incorrect?

2/ I dont understand why Bunuel subtract 324700 from 458600. What does 133900 refer to?

3/ I found this type of question quite challenging. Can anyone post the links to similar gmat problems in which the same concept is tested?

Thank you very much!

Hi,
you are also doing the same thing what has been done in prior solution..
we have to find numbers that have 13 in the end..
324700 will have 324713..
324800 will have 324813..

so The number of 100s from 324700 to 458600 is found by subtrcting these two and then dividing by 100..
so difference in numbers= 458600-324700=133900..
How many 100s does it contain 133900/100=1339
each of these 1339 100s has one 13 in it so ans 1339
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Re: How many integers between 324,700 and 458,600 have tens  [#permalink]

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VeritasPrepKarishma wrote:
144144 wrote:
is there another way to solve this?

I once saw someone solving this kind of questions by multiplying the amount of different number of digits u can put in each place (in each different digit)

You can but you would have to take multiple cases and that would be really cumbersome. The left most digit can be either a 3 or a 4 but then next digit depends on what the leftmost digit is. If left most digit is 3, next digit can be anything from 2 to 9. If the leftmost digit is 4, the next digit can be from 0 to 5. Similarly other digits too.
So preferably, focus on the approach given by Bunuel.

Hi!
I did it this way. Though i am getting the same answer, is it a correct approach?

For, 713 - 1613
1 x 10 x 1 x 1 = 10

Now 458-324 = 134.

Therefor total = 134 x 10 =1340

Total final possibilities = 1340 - 1 (1613 is not possible) = 1339 Re: How many integers between 324,700 and 458,600 have tens   [#permalink] 10 Aug 2017, 03:46

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