Is 1/p r/(r^2 + 2) ? (1) p = r (2) r 0

Question

Is  \frac{1}{p} > \frac{r}{r^2 + 2}  ?

(1)  p = r

(2)  r > 0

1/p > r /r^2 +2
the question is asking:
 r^2 +2 > pr ??

From stmt 1 we know:
r^2 +2 > r^2 ..so why s the answer not A

Bunuel · Accepted Answer

Is  \frac{1}{p}>\frac{r}{r^2 + 2} ?

(1)  p=r :

The question becomes:

Is  \frac{1}{r}>\frac{r}{r^2+2}?

Since  r^2+2  is always positive, multiplying inequality by this expression we'll get:

Is  \frac{r^2+2}{r}>r?

Is  r+\frac{2}{r}>r?

Is  \frac{2}{r}>0?

This inequality is true when  r>0  and not true when  r<0 . Not sufficient.

(2)  r>0 . Not sufficient by itself.

(1)+(2) Since from (2)  r>0 , then  \frac{2}{r}>0   IS  true and we have an YES answer to the question. Sufficient.

Answer: C.

tejal777 again when you simplifying  \frac{1}{p} >\frac{r}{r^2 +2}  to  r^2 +2 > pr , you are making a mistake. You  can  multiply inequality by r^2 +2 as this expression is always positive, BUT you  cannot  multiply inequality by p, as we don't know the sigh of p.

Hope it's clear.

whiplash2411 · Answer

The question can be rephrased as: p < \frac{r^2+2}{r} , by taking the reciprocals on each side (The inequality sign switches here, note that!) Statement 1: p = r This means the RHS becomes: \frac{p^2 + 2}{p} = p + \frac{2}{p} . Note that p + \frac{2}{p} will be greater than p (you're adding the term 2/p to it) as long as p > 0. If p < 0, then ultimately, you're subtracting a number from p, and hence it'll be lesser. Since there are two cases here, this is insufficient by itself. So we are down to B, C and E. Statement 2: r > 0 Does this say anything at all about the relationship between p and r? No. Insufficient. So, if you combine the statements, then p = r and r > 0, which means we've answered the only condition we had while solving the first statement and hence the statements together are sufficient. Hence the answer is C. Hope this helps.

tejal777 · Answer

Great wayof approaching!Super clear..thanks!

tripatsu · Answer

Thanks Bunuel , your approach to the problem was very good. !

thanks

mikemcgarry · Answer

I'm happy to help with this one.

Is (1/x) > y/(y^2+3) (1) y > 0 (2) x = y Statement #1 : y > 0 That tells us no information about x, so it's not sufficient by itself. Statement #2 : x = y Let's say x = y = 1. Then the left is 1, the right side is 1/(1^2 + 3) = 1/4, and the left side is bigger. BUT, if x = y = -1, then the left side is -1, and the right side is -1/4, and --- here's one of the really tricky things about negatives and inequalities --- the " less negative" number -1/4 is greater than -1, so the right side is bigger. It may be less confusing to think about that in terms of whole numbers ---- for example, 10 > 5, but -5 > -10: it's better to have $10 in your pocket rather than $5 in your pocket, but it's better to be $5 in debt than $10 in debt. Does that make sense? You are perfectly right --- y^2 is positive whether y is positive or negative, and therefore the denominator (y^2 + 3) is the same whether y is positive or negative, but what's different are whether the fractions themselves are negative, and that's what can reverse the order of the inequality. Without knowing whether x & y are positive and negative, we cannot determine the direction of the inequality. Statement #2 by itself is not sufficient Combined : y > 0 AND x = y Now, we are guaranteed that the fractions are both positive, so multiplying by x or y will not reverse the order of the inequality. Because x = y, we have (1/y) > y/(y^2+3). Cross-multiplying, we get y^2 + 3 > y^2, which is always true. Together, the statements are sufficient. Answer choice = C. Does that make sense? Please let me know if you have any further questions. Mike :-)

mohnish104 · Answer

mr bunuel how does r^2+2/r >r => r^2>0. shouldnt it be r^2+ 2 >r^2 instead

Bunuel · Answer

\frac{1}{r}>\frac{r}{r^2+2}?  --> as  r^2+2  is always positive, multiplying inequality by this expression we'll get:  \frac{r^2+2}{r}>r?  -->  r+\frac{2}{r}>r?  -->  \frac{2}{r}>0? .

As for your solution: we cannot cross-multiply  \frac{1}{r}>\frac{r}{r^2+2}  since we don't know whether r is positive or negative.

Hope it's clear.

architbansal · Answer

1/p > r/r^2+2 
=> r^2 + 2 > pr
=> r^2 - pr > -2
=> r(r - p) > -2
now if p=r than r-p becomes 0 
so => 0>-2 true

Why not statement A is suffiecient

Bunuel · Answer

We cannot cross-multiply in this case.

We  can  multiply 1/p > r/(r^2+2) by r^2+2 because r^2+2=non-negative+positive=positive but we  cannot  multiply 1/p > r/(r^2+2) by p because we don't know its sign:
If p is positive, then we'd get 1 > p*r/(r^2+2): keep the sign when multiplying by positive value; 
If p is negative, then we'd get 1  <  p*r/(r^2+2): flip the sign of the inequality when multiplying by negative value.

Hope it helps.

BrentGMATPrepNow · Answer

Target question :    Is 1/p > r/(r² + 2) ?

Statement 1 : p = r  
Take the target question and replace p with r to get:  Is 1/r > r/(r² + 2) ? 
Since r² + 2 is always POSITIVE, we can safely multiply both side of the inequality by r² + 2. 
When we do this, we get:  Is (r² + 2)/r > r ?

Now we can apply the following fraction property:   (a + b)/c = a/c + b/c  
We get:  Is r²/r + 2/r > r ? 
Simplify to get:  Is r + 2/r > r ? 
Subtract r from both sides to get:   Is 2/r > 0 ?  
At this point, we can see that statement 1 is not sufficient. 
When r is POSITIVE, the answer to our revised target question is  YES, it IS the case that 2/r > 0 
When r is NEGATIVE, the answer to our revised target question is  NO, it is NOT the case that 2/r > 0 
Since we cannot answer the  target question  with certainty, statement 1 is NOT SUFFICIENT

Statement 2 : r > 0  
Since we have no information about p, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined    
From statement 1 we learned that . . . 
When r is POSITIVE, the answer to our revised target question is  YES, it IS the case that 2/r > 0 
When r is NEGATIVE, the answer to our revised target question is  NO, it is NOT the case that 2/r > 0

Statement 2 tells us that r is POSITIVE
So, we can be certain that the answer to our revised target question is  YES, it IS the case that 2/r > 0 
Since we can answer the  target question  with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent

Gprabhumir · Answer

1) p=r

Then,
1/p > r/(r^2+2)
Multiplying p=r on both sides,
1>r^2/(r^2+2)
Square are always positive.
As x/(x+something) <1, for any positive integer x, should the answer not be A instead of C???

Can someone please explain.

Flower_Child · Answer

You can remember the following rule -
 Whenever you multiply or divide an inequality by a negative number, you must flip the inequality sign.

In this scenario :
 \frac{1}{r} > \frac{r}{r^2+2} 
we can't cross multiply r from  \frac{1}{r}   as we don't know the sign of r . No matter whether p=r ,we need to know the sign to cross multiply.

Here we can either multiply  {r^2+2} . Or simplify like below :

\frac{1}{r} - \frac{r}{r^2+2} >0

manutdtillidie · Answer

statement 1 P=r

let p=1 r=1

1/1>1/1+2 = 1>1/3 ------ yes

let p=-1 r=-1

-1/1>-1/3 ---------no 
statement 1 insufficient

statement - 2 r>0 
let r= 1
1/p>1/3 
no information about p hence statement 2 insufficient

combining 1 & 2

r=1 p=1

1/1>1/3----yes

r=2 p=2

1/2>2/6------yes

answer is C

Bunuel  is my approach correct ?

kelly_jacques · Answer

Bunuel I understand the logic of not cross multiplying given you don't know the value of r. However, I cross multiplied and just did two scenarios (1) if r was positive (2) if r was negative and deemed first question insufficient. The second question tells you its negative, so I can select the correct equation of the two and answer together is sufficient.

My confusion is just that cross-multiplying doesn't give the same formula. It gives 0>-2. or 0<-2 Is that a problem ? Is my method still a mistake?

My confusion is just that cross-multiplying doesn't give the same formula. It gives 0>-2. or 0<-2 Is that a problem ? Is my method still a mistake?

Bunuel · Answer

Your question is a bit challenging to follow entirely, but here's my response: Firstly, it's crucial to understand that during cross-multiplication, we multiply by the denominators of the fractions. So, when cross-multiplying \frac{1}{p} > \frac{r}{r^2 + 2} , we multiply by r^2 + 2 and p, not by r. Secondly, if p is positive, after cross-multiplication (and keeping the sign as is), the question evolves to " is r^2 +2 > pr ? ". Conversely, if p is negative, then after cross-multiplication (and flipping the sign), the question becomes " is r^2 +2 < pr ? ". For (1) that says p = r, if p (or r, since they're equal) is positive, the question simplifies to " is 2 > 0 ? ", hence the answer would be YES. For negative p, the question simplifies to " is 2 < 0 ? ", thus the answer would be NO. However, as we lack information about the sign of p (or r), we cannot say which case we have. . (2) is obviously useless alone, but when paired with (1), it supplies the missing information: r, and thereby, p is positive. So, the question simplifies to " is 2 > 0 ? " to which the answer is clearly YES. Generally, you should never multiply an inequality by a variable when its sign is unknown.

Robo_123 · Answer

why is the answer not A?
1/p-r/r^2+2>0
r^2+2-pr>0
r^2-pr>-2
p=r
r^2-r^2 = 0 > -2

Bunuel · Answer

Please check this post: https://gmatclub.com/forum/is-1-p-r-r-2 ... l#p3224302 P.S. Pure algebraic questions are no longer a part of the DS syllabus of the GMAT. DS questions in GMAT Focus encompass various types of word problems, such as: Word Problems Work Problems Distance Problems Mixture Problems Percent and Interest Problems Overlapping Sets Problems Statistics Problems Combination and Probability Problems While these questions may involve or necessitate knowledge of algebra, arithmetic, inequalities, etc., they will always be presented in the form of word problems. You won’t encounter pure "algebra" questions like, "Is x > y?" or "A positive integer n has two prime factors..." Check GMAT Syllabus for Focus Edition You can also visit the Data Sufficiency forum and filter questions by OG 2024-2025, GMAT Prep (Focus), and Data Insights Review 2024-2025 sources to see the types of questions currently tested on the GMAT. So, you can ignore this question. Hope it helps.

Is 1/p > r/(r^2 + 2) ? (1) p = r (2) r > 0

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