Data Sufficiency

Hi Marcelo,

Please post only one question per thread. Otherwise things can become pretty complicated when there are discussions on multiple questions.

Cheers,
Brent

In the xy-plane, does the line with equation y = 3x + 2 contain the point (r,s) ?

1. (3r + 2 - s) (4r + 9 - s) = 0
2. (4r - 6 - s) (3r + 2 - s) = 0

Target question: Does the line with equation y = 3x + 2 contain the point (r,s)

If (r,s) is on the line defined by the equation y = 3x + 2, then (r,s) must SATISFY the equation y = 3x + 2. In other words, it must be true that s = 3r + 2
For example: We know that the point (5, 17) is on the line y = 3x + 2, because when we plug x = 5 and y = 17 into the equation, we get 17 = 3(5) + 2 and the equation HOLDS TRUE.

So, we can REPHRASE the target question as "Does s = 3r + 2?"

Statement 1: (3r+2-s)(4r+9-s) = 0
From this, we know that EITHER (3r+2-s) = 0 OR (4r+9-s) = 0
If (3r+2-s) = 0 then s = 3r+2, in which case the answer to our REPHRASED target question is yes
If (4r+9-s) = 0 then s = 4r+9, in which case the answer to our REPHRASED target question is no
Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: (4r-6-s)(3r+2-s) = 0
From this, we know that either (4r-6-s) = 0 or (3r+2-s) = 0
If (4r-6-s)) = 0 then s = 4r-6, in which case the answer to our REPHRASED target question is no
If (3r+2-s) = 0 then s = 3r+2, in which case the answer to our REPHRASED target question is yes
Since we cannot answer the REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Since (3r+2-s) is the only expression common to BOTH equations, it MUST be true that 3r+2-s = 0, in which case s MUST equal 3r+2
Since we can answer the REPHRASED target question with certainty, the combined statements are SUFFICIENT

Answer = C


Cheers,
Brent

Aside: We have a free video with tips on rephrasing the target question: https://www.gmatprepnow.com/module/gmat- ... cy?id=1100

What fraction of this year's graduation students at a certain college are males?

(1) Of this year's graduation students, 33% of male and 20% of female transferred from another college.
(2) Of this year's graduation students, 25% transferred from another college.

Here's one approach.

Let M = ALL male graduates
Let F = ALL female graduates

Target question: What fraction of this year's graduation students at a certain college are male?
In other words, we want the value of M/(M+F), so we can rephrase the target question...

REPHRASED target question: What is the value of M/(M+F)?

Statement 1:Of this year's graduation students, 33% of male and 20% of female transferred from another college.
In other words, the TOTAL number of graduates who transferred = 0.33M + 0.2F
This info is not enough to find the value of M/(M+F)
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: Of this year's graduation students, 25% transferred from another college.
Total number of graduates who transferred = 0.25(M+F)
This info is not enough to find the value of M/(M+F)
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that the total number of graduates who transferred = 0.33M + 0.2F
Statement 2 tells us that the total number of graduates who transferred = 0.25(M+F)
So, we can conclude that 0.33M + 0.2F = 0.25(M+F)
Expand: 0.33M + 0.2F = 0.25M + 0.25F
Rearrange to get: 0.08M = 0.05F (Perfect)
Multiply both sides by 100 to get 8M = 5F
So, F = 8M/5
From here, we can find the value of M/(M+F)
Since 8M/5 = F, we can replace F with 8M/5 to get:
M/(M + F) = M/(M + 8M/5)
= M/(13M/5)
= 5M/13M
= 5/13
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer = C

Cheers,
Brent

If p and n are positive integers and p > n, what is the remainder when p² - n² is divided by 15?
(1) The remainder when (p + n) is divided by 5 is 1.
(2) The remainder when (p - n) is divided by 3 is 1.

Target question: What is the remainder when p² - n² is divided by 15

NOTE that p² - n² is a difference of squares, so we can factor it to get: p² - n² = (p + n)(p - n). Since both (p + n) and (p - n) are in the statements, it may be useful to REPHRASE the target question...

REPHRASED target question: What is the remainder when (p + n)(p - n) is divided by 15?

Statement 1: The remainder when (p + n) is divided by 5 is 1
This tell us that (p + n) is NOT DIVISIBLE by 5.
Since there's no information about (p-n), we can't determine the remainder when (p + n)(p - n) is divided by 15

Consider these two conflicting cases:
Case a: p = 5 and n = 1 (notice that the remainder when p+n is divided by 5 is 1). In this case, the remainder when is 9 when (p + n)(p - n) is divided by 15
Case b: p = 1 and n = 0 (notice that the remainder when p+n is divided by 5 is 1). In this case, the remainder when is 1 when (p + n)(p - n) is divided by 15
So, 2 of the numbers are less than 30
Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: The remainder when p - n is divided by 3 is 1
Here we have no information about p+n.
Consider these two conflicting cases:
Case a: p = 5 and n = 1 (notice that the remainder when p-n is divided by 3 is 1). In this case, the remainder when is 9 when (p + n)(p - n) is divided by 15
Case b: p = 1 and n = 0 (notice that the remainder when p-n is divided by 3 is 1). In this case, the remainder when is 1 when (p + n)(p - n) is divided by 15
So, 2 of the numbers are less than 30
Since we cannot answer the REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
IMPORTANT: Notice that I happened to use the same values for the counter-examples in each statement. This means that we can use the same values here to show that the COMBINED statements are not sufficient. That is...
Consider these two conflicting cases:
Case a: p = 5 and n = 1 (notice that both statements are satisfied). In this case, the remainder when is 9 when (p + n)(p - n) is divided by 15
Case b: p = 1 and n = 0 (notice that both statements are satisfied). In this case, the remainder when is 1 when (p + n)(p - n) is divided by 15
So, 2 of the numbers are less than 30
Since we cannot answer the REPHRASED target question with certainty, the COMBINED statements are NOT SUFFICIENT

Answer: E

ALTERNATIVELY, when examining the statements combined, we can use a nice rule that says:
If N divided by D, leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.
For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, 1+5, 1+(2)(5), 1+(3)(5), 1+(4)(5), . . . etc.

Okay, onto the question . . .
Statement 1: Applying the above rule, some possible values of p+n are 6, 11, 16, 21, 26, etc.
Aside: you'll notice that I didn't include 1 as a possible value since we're told that p and n are positive integers, and we can't get a sum of 1 if both are positive

Statement 2: Applying the above rule, some possible values of p-n are 1, 4, 7, 10, 13, etc

Let's examine two cases with conflicting results.

case a: p+n = 11 and p-n = 1
Add the equations to get 2p = 12, which means p = 6 and n = 5 (perfect, we have positive integer values for p and n)
In this case, when (p + n)(p - n) is divided by 15, the remainder is 11

case b: p+n = 6 and p-n = 4
Add the equations to get 2p = 10, which means p = 5 and n = 1 (perfect, we have positive integer values for p and n)
In this case, when (p + n)(p - n) is divided by 15, the remainder is 9
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Answer = E

Cheers,
Brent

Are x and y both positive?
1) 2x - 2y =1
2) x/y >1

Target question: Are x and y both positive?

Statement 1: 2x - 2y = 1
There are several pairs of numbers that satisfy this condition. Here are two:
Case a: x = 1 and y = 0.5, in which case x and y are both positive
Case b: x = -0.5 and y = -1, in which case x and y are not both positive
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x/y > 1
This tells us that x/y is positive. This means that either x and y are both positive or x and y are both negative. Here are two possible cases:
Case a: x = 4 and y = 2, in which case x and y are both positive
Case b: x = -4 and y = -2, in which case x and y are not both positive
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2
Statement 1 tells us that 2x - 2y = 1.
Divide both sides by 2 to get: x - y = 1/2
Solve for x to get x = y + 1/2

Now take the statement 2 inequality (x/y > 1) and replace x with y + 1/2 to get:
(y + 1/2)/y > 1
Rewrite as: y/y + (1/2)/y > 1
Simplify: 1 + 1/(2y) > 1
Subtract 1 from both sides: 1/(2y) > 0
If 1/(2y) is positive, then y must be positive.

Statement 2 tells us that either x and y are both positive or x and y are both negative.
Now that we know that y is positive, it must be the case that x and y are both positive
Since we can now answer the target question with certainty, the combined statements are SUFFICIENT

Answer = C

Cheers,
Brent

confused13 wrote:Question on last DS Problem

x-y=1/2

e.g.

on stmt A:
x=2 y=1.5 --> YES
x=0 Y=-0.5 --> NO
--> not suficient

stmt B says only x>y
which is clearly no sufficient

but given my above examples which also fullfill statement 2, I don't know why the answer is C and NOT E.

x=0 Y=-0.5 does not satisfy statement 2.
x/y = 0/(-0.5) = 0, and 0 is not greater than 1.

Cheers,
Brent