Inequality

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nysnowboard: Junior | Next Rank: 30 Posts; Posts: 23; Joined: Sat Mar 06, 2010 6:07 am; Thanked: 1 times

by nysnowboard » Thu May 20, 2010 10:23 am

3. If 5 < x < 10 and y = x + 5, what is the greatest possible integer value of x + y ?
(A) 18
(B) 20
(C) 23
(D) 24
(E) 25

I approached the solution by saying the maximum integer value of x is 9.
Hence, maximum value of y is y=x+5=9+5=14

So the maximum integer value of x+y = max x + max y = 9 +14 = 23
I know I am missing something but could someone point out my logic error?

*EDIT* While posting this problem I realized my error. While the question asks for the greatest integer value of x + y, it doesn't state that x or y need to be integers themselves. So, x could be 9.9, which would allow an integer of 24. Careless assumption on my part. I will post this nonetheless in the hopes it may keep someone else from making the same error.

Below is the solution:

[spoiler]3. Adding x to both sides of the equation y = x + 5 yields x + y = x + (x + 5), or x + y = 2x + 5. Hence, the
greatest possible value of x + y is the maximum possible value of 2x + 5. Now, let's create this expression
out of the given inequality 5 < x < 10. Multiplying the inequality by 2 yields 10 < 2x < 20. Adding 5 to each
part of the inequality yields 10 + 5 < 2x + 5 < 20 + 5, or 15 < 2x + 5 < 25. So, 2x + 5 is less than 25. The
greatest possible integer value of 2x + 5 is 24. Hence, the answer is (D).[/spoiler]

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Source: Beat The GMAT — Problem Solving | Thu May 20, 2010 10:23 am

debmalya_dutta: Master | Next Rank: 500 Posts; Posts: 385; Joined: Sun Jul 12, 2009 10:16 pmFollowed by: 2 members; Thanked: 29 times; GMAT Score:710

by debmalya_dutta » Thu May 20, 2010 11:08 am

since
5<x<10 say this is inequality # 1
adding 5 to all the sides of the inequality
10<x+5<15 which is 10<y<15 say this is inequality # 2
Adding inequality # 1 and inequality # 2
15< x+y < 25
since x+y is an integer and has to be less than 25, the max integer value of x+y =24

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Stuart@KaplanGMAT: GMAT Instructor; Posts: 3225; Joined: Tue Jan 08, 2008 2:40 pmFollowed by: 614 members; Location: Toronto; Thanked: 1710 times; GMAT Score:800

by Stuart@KaplanGMAT » Thu May 20, 2010 1:13 pm

nysnowboard wrote:3. If 5 < x < 10 and y = x + 5, what is the greatest possible integer value of x + y ?
(A) 18
(B) 20
(C) 23
(D) 24
(E) 25

*EDIT* While posting this problem I realized my error. While the question asks for the greatest integer value of x + y, it doesn't state that x or y need to be integers themselves. So, x could be 9.9, which would allow an integer of 24. Careless assumption on my part. I will post this nonetheless in the hopes it may keep someone else from making the same error.

Below is the solution:

Quick note: since y = x+5, the only way that x + you could be an integer is if x and y are both integers or they both have ".5" as their decimals. The example you gave, 9.9, would generate y=14.9, and that wouldn't give an integer value for x+y.

So, to maximize the integer value of x+y, we choose x=9.5 and that makes y=14.5.

Stuart Kovinsky | Kaplan GMAT Faculty | Toronto

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nysnowboard: Junior | Next Rank: 30 Posts; Posts: 23; Joined: Sat Mar 06, 2010 6:07 am; Thanked: 1 times

by nysnowboard » Thu May 20, 2010 1:21 pm

Thanks for pointing that out. I realized that if x +y could reach a value of 9.9 + 14.9 = 24.8 (also note that it can come infinitely close to but will never reach 25)x +y had to pass the integer value of 24 along the way... unfortunately my mind isn't as elegant and logical as of those of the experts here... once it sees a light at the end of the tunnel, it rushes for it regardless of the outcome.

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frank1: Legendary Member; Posts: 537; Joined: Fri Jan 22, 2010 10:06 pmFollowed by: 1 member; Thanked: 14 times

by frank1 » Thu May 20, 2010 8:49 pm

nysnowboard wrote:Thanks for pointing that out. I realized that if x +y could reach a value of 9.9 + 14.9 = 24.8 (also note that it can come infinitely close to but will never reach 25)x +y had to pass the integer value of 24 along the way... unfortunately my mind isn't as elegant and logical as of those of the experts here... once it sees a light at the end of the tunnel, it rushes for it regardless of the outcome.

i dont think its your fault at all...
it is human tendency and actually the question has been setup so that student fall in that pit (agree or not that is one major intension of test question writers)......
So,they setup a trap here and many falls....it is natural human tendency....
even i thought 23 at first
see how the word 'integer' is given at last....(somebody said me that gmac invest lots of money to setp up trap in questions...lol)
but i thought if it was 23 it wont be 2 minute question and gmac may not ask me simple addition ...lol....so i relooked into the question
infact the question turn out very diffucult then i started thinking like
9.9+14.9
9.8+14.2
...
but setting up enquility was much easier.....

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kstv: Legendary Member; Posts: 610; Joined: Fri Jan 15, 2010 12:33 amFollowed by: 2 members; Thanked: 47 times

by kstv » Thu May 20, 2010 9:57 pm

If 5 < x < 10 and y = x + 5, what is the greatest possible integer value of x + y ?
(A) 18
(B) 20
(C) 23
(D) 24
(E) 25

y = x+5 adding x on both sides
y+x = 2x+5
5<x<10
to maximise y+x which has to be an integer 2x+5 has to be an integer.
if x is 9.5 2x is 19, y+x = 24
y+x = 25 only if x is 10 but x < 10
so IMO D

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quantskillsgmat: Senior | Next Rank: 100 Posts; Posts: 96; Joined: Fri Apr 23, 2010 1:14 amFollowed by: 1 member; Thanked: 1 times

by quantskillsgmat » Fri May 21, 2010 12:34 am

dont get confused that x and y are integers.since you need greatest integral values of x+y where x and y can be decimals also.
lets take x=9.9 so y=5+9.9=14.9
so x+y=9.9+14.9=24.8 so greatest integral value of x+y is 24.
quantskills
wisdommart delhi

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analyst218: Master | Next Rank: 500 Posts; Posts: 140; Joined: Fri Feb 05, 2010 2:43 pm; Thanked: 3 times; GMAT Score:720

by analyst218 » Tue May 25, 2010 2:28 pm

nysnowboard wrote:3. If 5 < x < 10 and y = x + 5, what is the greatest possible integer value of x + y ?
(A) 18
(B) 20
(C) 23
(D) 24
(E) 25

I approached the solution by saying the maximum integer value of x is 9.
Hence, maximum value of y is y=x+5=9+5=14

So the maximum integer value of x+y = max x + max y = 9 +14 = 23
I know I am missing something but could someone point out my logic error?

*EDIT* While posting this problem I realized my error. While the question asks for the greatest integer value of x + y, it doesn't state that x or y need to be integers themselves. So, x could be 9.9, which would allow an integer of 24. Careless assumption on my part. I will post this nonetheless in the hopes it may keep someone else from making the same error.

Below is the solution:

[spoiler]3. Adding x to both sides of the equation y = x + 5 yields x + y = x + (x + 5), or x + y = 2x + 5. Hence, the
greatest possible value of x + y is the maximum possible value of 2x + 5. Now, let's create this expression
out of the given inequality 5 < x < 10. Multiplying the inequality by 2 yields 10 < 2x < 20. Adding 5 to each
part of the inequality yields 10 + 5 < 2x + 5 < 20 + 5, or 15 < 2x + 5 < 25. So, 2x + 5 is less than 25. The
greatest possible integer value of 2x + 5 is 24. Hence, the answer is (D).[/spoiler]

well since you know that maximum for X is 9, the sum could be 23 for sure.
on a second look, you can find that x can be decimals, and since y=x+5, the only decimal
that will give an integer when added, is 0.5 ..

so the answer is +1 to 23, which is 24.

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