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# Two positive integers differ by 4, and sum of their reciprocals is 10/

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Manager
Joined: 08 Oct 2010
Posts: 194
Location: Uzbekistan
Schools: Johnson, Fuqua, Simon, Mendoza
WE 3: 10
Two positive integers differ by 4, and sum of their reciprocals is 10/  [#permalink]

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26 Oct 2010, 02:40
3
00:00

Difficulty:

25% (medium)

Question Stats:

73% (01:32) correct 27% (02:35) wrong based on 98 sessions

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Two positive integers differ by 4, and sum of their reciprocals is 10/21. Then one of the numbers is

a) 3
b) 1
c) 5
d) 21
e) 28
Manager
Joined: 02 Apr 2010
Posts: 98
Re: Two positive integers differ by 4, and sum of their reciprocals is 10/  [#permalink]

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26 Oct 2010, 03:53
There are several ways to solve this question:

Algebraic approach:

Let n be the smaller integer => 1/n + 1/(n+4) = 10/21
Solve for n => n=3. Hence, answer b) is correct.

Picking numbers:

5 obviously cannot be the larger of the two integers as this would mean that the smaller integer would equal 1 (hence reciprocal also 1) and the sum of the reciprocals would exceed 10/21.
If 5 is the smaller of the two integers => 1/5 + 1/9 = 14/45. This is smaller than 10/21.

Next, choose 3 and 7 (because 12 and 28 would obviously make sum of the reciprocals too small) => 1/3 + 1/7 = 10/21. Hence, answer b) is correct.

Considering Least common multiple:

For the sum of the reciprocals of the integers to have the denominator 21 both numbers must be divisors of 21. This only holds true for answer choice b).
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 9330
Location: Pune, India
Re: Two positive integers differ by 4, and sum of their reciprocals is 10/  [#permalink]

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26 Oct 2010, 04:23
My take on this question - to solve it in the least time - is to make algebraic equation and then pick numbers.

If one number is n, the other is (n + 4).
Given 1/n + 1/(n+4) = 10/21
I look at the options and say that I need 21 as the denominator. 3 is given as an option. If 3 is one number, the other will be 7. Quickly substitute the values and confirm.

1/3 + 1/7 = 10/21

Note: In GMAT it is advisable to confirm your answer even if it seems pretty obvious. They like to put small little tricks to waylay you!
_________________
Karishma
Veritas Prep GMAT Instructor

Manager
Joined: 25 Aug 2010
Posts: 62
Re: Two positive integers differ by 4, and sum of their reciprocals is 10/  [#permalink]

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27 Oct 2010, 02:48
1/n + 1/(n+4) = 1/21 =====> n = 3
Senior Manager
Joined: 12 Sep 2017
Posts: 267
Re: Two positive integers differ by 4, and sum of their reciprocals is 10/  [#permalink]

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08 Jan 2019, 19:06
Hello!

Could someone please solve by the algebraic approach?

Kind regards!
Re: Two positive integers differ by 4, and sum of their reciprocals is 10/   [#permalink] 08 Jan 2019, 19:06
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