Data Sufficiency for 780+ Aspirants

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sureshbala: Master | Next Rank: 500 Posts; Posts: 319; Joined: Wed Feb 04, 2009 10:32 amFollowed by: 9 members; Location: Delhi; Thanked: 84 times

by sureshbala » Tue Feb 17, 2009 12:07 pm

Folks, in this thread we will get into DS practice questions on a regular basis. I will come up with questions keeping in mind the standard of the questions that one needs to be comfortable with in order to score top in Quant Section of the exam.

Here are the two simple rules that I want all the members to strictly adhere to..

1. Do not post your questions in this thread. Instead start a new thread or post in the relevant thread.

2. Please give a bit of explanation for the answer that you post here.(Don't simply post the answer)

Here is our first question.

X and Y are real numbers such that X/Y > 2. Is 3X + 2Y <17?

I. X - Y < 2

II. Y - X < 2

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Source: Beat The GMAT — Data Sufficiency | Tue Feb 17, 2009 12:07 pm

bluementor: Master | Next Rank: 500 Posts; Posts: 418; Joined: Wed Jun 11, 2008 5:29 am; Thanked: 65 times

Re: Data Sufficiency for 780+ Aspirants

by bluementor » Tue Feb 17, 2009 1:43 pm

sureshbala wrote: Here is our first question.

X and Y are real numbers such that X/Y > 2. Is 3X + 2Y <17?

I. X - Y < 2

II. Y - X < 2

x/y > 2 would mean that either both x and y are positive or both x and y are negative. If both x and y are negative, then the answer to the question stem is YES. So only the positive values of x and y need to be considered in trying to prove 3x + 2y<17 to be false.

Statement 1:
x-y < 2

For both x and y being positive, they need to be 'close' enough to zero in order to preserve x/y>2 and x-y<2. Therefore if x and y are small, 3x + 2y will be less than 17.

For eg. x = 3, y=1.1, 3x + 2y = 11.2 < 17. Other possible values of x and y will not be too far from those in this example and will satisfy 3x+2y<17. Therefore, statement 1 is sufficient.

Statement 2:
y-x < 2

If x=10, y=1, then 3x+2y will be larger than 17. Therefore insufficient.

Choose A.

P.S. My solution is a bit crude, especially in solving for statement 1. Would be interested in seeing a more analytical approach.

-BM-

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4meonly: Legendary Member; Posts: 891; Joined: Sat Aug 16, 2008 4:21 amFollowed by: 1 member; Thanked: 27 times; GMAT Score:660(

by 4meonly » Tue Feb 17, 2009 2:50 pm

I got B

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x2suresh: Master | Next Rank: 500 Posts; Posts: 258; Joined: Thu Aug 07, 2008 5:32 am; Thanked: 16 times

Re: Data Sufficiency for 780+ Aspirants

by x2suresh » Tue Feb 17, 2009 11:27 pm

sureshbala wrote:Folks, in this thread we will get into DS practice questions on a regular basis. I will come up with questions keeping in mind the standard of the questions that one needs to be comfortable with in order to score top in Quant Section of the exam.

Here are the two simple rules that I want all the members to strictly adhere to..

1. Do not post your questions in this thread. Instead start a new thread or post in the relevant thread.

2. Please give a bit of explanation for the answer that you post here.(Don't simply post the answer)

Here is our first question.

X and Y are real numbers such that X/Y > 2. Is 3X + 2Y <17?

I. X - Y < 2

II. Y - X < 2

X/Y >2 -- > both X and Y must be +ve or both -ve

3X + 2Y <17 always true when X and Y are -ve..

1) X - Y < 2 --> eqn(1)

X/Y >2 --> X-2Y>0 --> 2y-x<0 eqn(2)
eqn(1)+eqn(2)
--> y<2 --> X<4

3X + 2Y < 3*4+2*2 <16

Statement 1 is sufficient

2) Y-X <2

WHEN X=2.1 Y=1

3X + 2Y <17 True

WHEN X=30.1 Y=15
3X + 2Y <17 False

not sufficient

A

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Bidisha_800: Senior | Next Rank: 100 Posts; Posts: 44; Joined: Sun Feb 15, 2009 7:37 pmFollowed by: 1 member; Thanked: 3 times

Re: Data Sufficiency for 780+ Aspirants

by Bidisha_800 » Tue Feb 17, 2009 11:55 pm

1) X - Y < 2 --> eqn(1)

X/Y >2 --> X-2Y>0 --> 2y-x<0 eqn(2)
eqn(1)+eqn(2)
--> y<2 --> X<4

can we really add inequalities to "solve" like equation ?

I solved this the following way :

3x+2y
=x+2(x+y)
=y{x/y+2(1+x/y)}

x/y >2
x/y = 2+ d (delta) where d >0

3x+2y = y(8+3d)

From (1)

x-y <2
x-y = 2-m (m>0)
y(x/y -1) = 2- m
y(2+d-1) = 2 -m

y = (2-m)/(d+1) <2 ---------(a)

y(d+1) = 2-m < 2
5y < 10 ..............................(b)

now 3x+2y = y(8+3d)
=y {5 + 3(d+1)}
= 5y + 3(2-m)
<10 + 6 -3m
<16 -3m

since m >0 so 3x+2y has to be less than 16

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sureshbala: Master | Next Rank: 500 Posts; Posts: 319; Joined: Wed Feb 04, 2009 10:32 amFollowed by: 9 members; Location: Delhi; Thanked: 84 times

by sureshbala » Wed Feb 18, 2009 10:09 pm

Folks, any take on this....I am looking for more solutions (lucid) before I move away from this....

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pbanavara: Master | Next Rank: 500 Posts; Posts: 279; Joined: Wed Sep 24, 2008 8:26 am; Location: Portland, OR; Thanked: 6 times

by pbanavara » Wed Feb 18, 2009 10:58 pm

x/y < 2 From stem

1 x- y < 2

For this to be true both x and y have to be negative. If that's the case then 3x+2y is always < 17 - Sufficient.

2. y-x < 2 ; x<2y : For this to be true x is +ve and y is -ve always .. hence insufficient.

Choose A.

I doubt this is correct - but gave it a shot.

- pradeep

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sureshbala: Master | Next Rank: 500 Posts; Posts: 319; Joined: Wed Feb 04, 2009 10:32 amFollowed by: 9 members; Location: Delhi; Thanked: 84 times

by sureshbala » Fri Feb 20, 2009 5:21 am

pbanavara wrote:x/y < 2 From stem

1 x- y < 2

For this to be true both x and y have to be negative. If that's the case then 3x+2y is always < 17 - Sufficient.

2. y-x < 2 ; x<2y : For this to be true x is +ve and y is -ve always .. hence insufficient.

Choose A.

I doubt this is correct - but gave it a shot.

- pradeep

I think you were too quick in concluding the bold statements above..Isn't it?

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pbanavara: Master | Next Rank: 500 Posts; Posts: 279; Joined: Wed Sep 24, 2008 8:26 am; Location: Portland, OR; Thanked: 6 times

by pbanavara » Fri Feb 20, 2009 8:52 am

sureshbala wrote:
pbanavara wrote:x/y < 2 From stem

1 x- y < 2

For this to be true both x and y have to be negative. If that's the case then 3x+2y is always < 17 - Sufficient.

2. y-x < 2 ; x<2y : For this to be true x is +ve and y is -ve always .. hence insufficient.

Choose A.

I doubt this is correct - but gave it a shot.

- pradeep
I think you were too quick in concluding the bold statements above..Isn't it?

So x-y < 2 and x > 2y or x< y+2 and x > 2y

Try x +ve and y +ve - doesn't hold true
Try x -Ve and y +ve - doesn't hold true
x +ve and y -ve - doesn't not true

x -ve and y -ve - Holds true. Unless am missing something here.

Hence the conclusion.

I had to edit the post multiple times .. darn this is what happens when work and GMAT prep clash.

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ven4gmat: Junior | Next Rank: 30 Posts; Posts: 29; Joined: Sun Feb 08, 2009 11:29 am; Thanked: 3 times

by ven4gmat » Sat Feb 21, 2009 10:06 am

I somehow managed to get the answer as A. But not confident enough... sureshbala, i hope you are coming up with the explanation

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sureshbala: Master | Next Rank: 500 Posts; Posts: 319; Joined: Wed Feb 04, 2009 10:32 amFollowed by: 9 members; Location: Delhi; Thanked: 84 times

by sureshbala » Mon Feb 23, 2009 5:47 am

Given x/y>2…..implies both x and y can be positive or both of them are negative
If both of them are negative, it is clear that 3x+2Y<0.
If both of them are positive we have x > 2y…......(1)

Statement1: x-y < 2 ...(2)
From (1) we have x-y > y .....(3)
From (2) and (3), y<x-y<2, so y<2.
From (2), we have x<y+2, so x<4.
Since y<2 and x<4, 3x+2y<16.
Hence statement (1) alone is sufficient

Coming to statement 2, we can think along the same lines and easily find two contradicting examples.
Let x=100 and y=1
Clearly x/y > 2, and y-x < 2 and the value of 3x+2y=302 (>18)
Let x=3 and y=1
Clearly x/y>2 and y-x<2 and the value of 3x+2y=11(<18)
So statement(2) is not sufficient
Hence the answer is Choice A.

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sureshbala: Master | Next Rank: 500 Posts; Posts: 319; Joined: Wed Feb 04, 2009 10:32 amFollowed by: 9 members; Location: Delhi; Thanked: 84 times

by sureshbala » Sun Mar 01, 2009 10:28 pm

Here is the next one...

The roots of a quadratic equation ax^2+bx+c=0 are integers and a+b+c>0 and a>0. Are both the roots of the equation positive?

1. Sum of the roots is positive.

2. Product of the roots is positive.

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Data Sufficiency for 780+ Aspirants

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Re: Data Sufficiency for 780+ Aspirants

by Stuart@KaplanGMAT » Mon Mar 02, 2009 12:24 am

sureshbala wrote:Here is our first question.

X and Y are real numbers such that X/Y > 2. Is 3X + 2Y <17?

I. X - Y < 2

II. Y - X < 2

A bit of thought and number picking is often the quickest way to solve high end DS questions. Remember, no bonus points on test day for elegant solutions!

Original: x and y have the same sign and x/y > 2. As noted, double negative values will give us a "yes" answer all the time, so we just need to try to get a "no" answer using positive values.

(1) x - y < 2

OK, now we need numbers that are less than 2 apart but for which x is more than double y. Well, if x=4 and y=2, then they're exactly 2 apart and x is exactly double y. If we pick numbers bigger than 4 and 2, then for x to be more than double y, it will have to be more than 2 greater than y. Therefore, x < 4 and y < 2. If x=4 and y=2 then we'd have:

3*4 + 2*2 = 16 which is less than 17, so x<4 and y<2 are definitely going to give us a "no" answer: sufficient.

(As an aside, if the question had been "Is 3x + 2y < 16" it would have been more interesting!)

(2) y - x < 2

Here we can quickly pick y = 1 and x = 100 to see that we can satisfy both rules and get a "yes" answer; since we already know that negatives are possible and that all negatives give us a "no" answer, insufficient.

(1) is suff, (2) isn't: choose (A).

This might seem time consuming, but that's only because I explained every step out loud. If you train yourself to think this way, this question is finished in under a minute.

Stuart Kovinsky | Kaplan GMAT Faculty | Toronto

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welcome: Master | Next Rank: 500 Posts; Posts: 137; Joined: Thu Jan 08, 2009 1:27 am; Thanked: 7 times

by welcome » Mon Mar 02, 2009 12:55 am

Dear Suesh,
Thanks for a good Qs and a nice answer.

Dear Stuart,
I made this question right by making 1st condition by inequalities calculations, but in condition 2 I did not had an option but for to choose nos.

Can you or Suresh, please explain us wha all we need to consider before choosing smart nos. for cheking the answers, if we are choosing not to go by algebric methods.

Thanks.

Shubham.
590 >> 630 >> 640 >> 610 >> 600 >> 640 >> 590 >> 640 >> 590 >> 590

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bluementor: Master | Next Rank: 500 Posts; Posts: 418; Joined: Wed Jun 11, 2008 5:29 am; Thanked: 65 times

by bluementor » Mon Mar 02, 2009 3:13 am

sureshbala wrote:Here is the next one...

The roots of a quadratic equation ax^2+bx+c=0 are integers and a+b+c>0 and a>0. Are both the roots of the equation positive?

1. Sum of the roots is positive.

2. Product of the roots is positive.

Basic rules:
a>0, this simply means the curve ax^2+bx+c=0 is 'U' shaped, having a single minimum point.
The sign of -b/2a can be used to identify if this single minimum point occurs in the positive x region (-b/2a > 0) or in the negative x region (-b/2a < 0).
The sign of c determines whether the curve intersects the y-axis in the positive y region (c>0) or the negative y region(c<0).

Statement 1: Sum of the roots is positive
This could mean either:
(i) both roots are positive (with c>0), or
(ii) one positive root and one negative root (with c<0, and the positive root having the larger absolute value)

For both these cases, the minimum point is in the positive x region. So we have:

-b/2a > 0
Since we know, a>0, therefore b must be less than 0.

a + b + c>0
We know that a>0 and b<0, but c could be either positive or negative. Therefore, we are unable to determine which of these two cases (i and ii) is true. Insufficient.

Statement 2: Product of the roots is positive
This could mean either:
(iii) both roots are positive (with -b/2a >0), or
(iv) both roots are negative (with -b/2a <0).

For both these cases, the y-intercept must be positive (c>0).

a + b + c>0
We know that a>0 and c>0, but b could be either positive or negative. Therefore, we are unable to determine which of these two cases (iii and iv) is true. Insufficient.

Both statements together:
From statement 1, we have established that b<0. This produces -b/2a > 0 (minimum point occuring in the positive x region)
From statement 2, we have established that c>0. The y-intercept in positive.

We can conclude now that both roots are positive.

Choose C.

-BM-

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