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# M01-22

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Math Expert
Joined: 02 Sep 2009
Posts: 52278

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15 Sep 2014, 23:15
19
00:00

Difficulty:

25% (medium)

Question Stats:

73% (01:11) correct 27% (01:34) wrong based on 159 sessions

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How many integers between 324,700 and 458,600 have a 2 in the tens digit and a 1 in the units digit?

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

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Math Expert
Joined: 02 Sep 2009
Posts: 52278

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15 Sep 2014, 23:15
7
5
Official Solution:

How many integers between 324,700 and 458,600 have a 2 in the tens digit and a 1 in the units digit?

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

Approach #1:

There is one number in each hundred with 2 in the 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 $$\frac{133,900}{100}=1,339$$ numbers.

Approach #2:

Look at the series of all numbers ending with 21 within the given range.
• 324,721
• 324,821
• .....
• 458,421
• 458,521

Following the series, we can see that only the first four numbers are changing. Instead of counting the numbers ending in 21 in the given range (324,700 to 458,600), simplify and only count the numbers in the range of 3,247 to 4,585. Therefore, the number of integers ending in 21 within the range is $$4,585-3,247+1 = 1,339$$.

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Intern
Joined: 21 May 2015
Posts: 15

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21 Aug 2017, 15:26
I think this is a high-quality question and I agree with explanation.
Intern
Joined: 10 Mar 2015
Posts: 14

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09 Mar 2018, 10:33
1
Check out the series below:
324,721
324,821
324,921
.....
458521

So, we could see that above series is in AP.
so, apply the formula and find out the number of terms.
first term=324,721
Common diff= 100
last term= 458521

458521=324721 + (n-1)100
this will give n=1339.
The right choice.

Please give kudos, if you really like my explanation.
Manager
Joined: 26 Feb 2018
Posts: 79
Location: United Arab Emirates
GMAT 1: 710 Q47 V41
GMAT 2: 770 Q49 V47

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12 Mar 2018, 12:39
I think the answer choices make this question a bit too easy.

The question asks "How many numbers that are xxx,x21" - obviously there would be one number for every 100, (and 10 for every complete 1000). So the real question is how many hundreds between the two numbers? You can immediately see that the difference between the hundreds digit is 9, and there is only one answer choice that ends in a 9. That means you can go for this one immediately without any calculation.

You can also simply subtract the two numbers which gives you 133900. This is 1339 * 100 and you have 1 number per 100

I feel a real GMAT question would throw in some "trip up" answers into the options, like 139 or 13390 or 1329, etc
Manager
Joined: 26 Feb 2018
Posts: 53
Location: India
GMAT 1: 560 Q41 V27
WE: Web Development (Computer Software)

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18 Apr 2018, 13:09
I think this is a high-quality question and I agree with explanation. This is a good quality question. The explanation provided is appropriate.
Intern
Joined: 09 Nov 2017
Posts: 6

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15 Oct 2018, 06:33
chetan2u, Bunuel

I was trying to solve this question by 324721 to 458521 .. where last 2 digits are fixed.
From 324721 to 399999 -- we can have only 8 (2 to 9) *6(4 to 9) * 3 (7 to 9) = 144 numbers ( as 3 and last 2 digits are fixed).
Similarly from 400000 to 458521 .. we can have 6(0 to 5 for ten thousands digit) * 9 * 6=324 ways.
so total 144+324 =468 ways.
Although i understood the answer for the question given... i did not understand through my above approach where am i goiing wrong. could you please help with this
Math Expert
Joined: 02 Aug 2009
Posts: 7200

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15 Oct 2018, 08:19
1
R999 wrote:
chetan2u, Bunuel

I was trying to solve this question by 324721 to 458521 .. where last 2 digits are fixed.
From 324721 to 399999 -- we can have only 8 (2 to 9) *6(4 to 9) * 3 (7 to 9) = 144 numbers ( as 3 and last 2 digits are fixed).
Similarly from 400000 to 458521 .. we can have 6(0 to 5 for ten thousands digit) * 9 * 6=324 ways.
so total 144+324 =468 ways.
Although i understood the answer for the question given... i did not understand through my above approach where am i goiing wrong. could you please help with this

Hi...
You miss out on a lot of combinations this way..
Just one example..
From 324721 to 399999 -- we can have only 8 (2 to 9) *6(4 to 9) * 3 (7 to 9) = 144 numbers ( as 3 and last 2 digits are fixed).
So you have taken hundreds place as only 7 to 9..
But if you have a number 325121.. this is a valid number but in your approach is not valid because of you taken hundreds as only 7, 8 or 9..
You could this by taking..
324700 to 324999....1*1*1*3*1*1=3
325000 to 329999....1*1*5*10*1*1=50
330000 to 399999..1* 7*10*10*1*1=700
400000 to 449999...1*5*10*10*1*1=500
450000 to 457999....1*1*8*10*1*1=80
458000 to 458600.. 1*1*1*6*1*1=6

Total ... 3+50+700+500+80+6=1339

A
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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Intern
Joined: 09 Nov 2017
Posts: 6

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15 Oct 2018, 09:30
chetan2u,

Thank you for such quick and clear reply
Re: M01-22 &nbs [#permalink] 15 Oct 2018, 09:30
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# M01-22

Moderators: chetan2u, Bunuel

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