Is t =|r - s|? (1) 3r 3s (2) t = r -s

Question

Is t =|r - s|?

(1) 3r > 3s

(2) t = r -s

carlosesg92 · Answer

First, we have to know that if r-s<0 so |r-s|=s-r and if r-s>=0; |r-s|=r-s

(1) 3r > 3s -> r-s>0 -> t=r-s but we don't have any value. Insufficient.

(2) t=r-s -> if is happens, so r-s must be >=0 to answer the question, but we don't have it in this point. Insufficient

Finally (1) and (2) are sufficient together, because if r-s>0, so t=r-s and we can confirm with the point (2).

Manat · Answer

@chetan2u:-can you please explain how this question needs to be solved?

ocelot22 · Answer

(1) Nothing about t NS

(2) t = r-s. It is tempting to mark sufficient, but lets test some values Let r=-2 t=-3 then t =-2--3 =1 and |r-s| - |1| =1 Yes

let r=-2, s=1 then t =-2 - 1 =-3 and |r-s| =|-3| = 3 NS

(1) and (2) 3r>3s --> r>s And t=r-s Logic tells us that t will always be positive thus suff, however we can test some values just to be sure.

Notice for values of r and s r >S So, let r=-3, s=-4 then r-s = -3 --4 = 1 and |r-s| = |1| =1 Yes

Let r=-1/2 and s=-1 then r-s =1/2 and |r-s| =1/2 Yes

Let r=-1/2 s = -3/4 then r-s = 1/4 and |r-s| = |1/4| = 1/4 Yes

Let r = 1/2 s =1/4 then r-s = 1/4 and |r-s| = 1/4 Yes

Let r=2 and s=1 then r-s = 2-1 = 1 and |r-s| =|1| = 1

So whether we pick negative integers, negative fractions, positive fractions, or positive integers for r and s we get a Yes answer we can mark sufficient

C

Gladiator59 · Answer

Manat , the question asks if t =|r-s|.. note that the modulus is always non-negative. So, we can answer this question as yes or no, based on knowing if t = r-s or t = s-r depending on which of r & s is bigger. For example, if r > s and t = r - s then we can say that yes, t = |r - s|, but it would be a certain "no" if r < s. Here, r - s would be negative and t cannot take a negative value. Let us look at - Statement (1) 3 r > 3s or, r > s ... does not give us any info to relate with t, hence insufficient Statement (2) t = r - s This too is insufficient as depending on which one is bigger, r - s could either be negative or positive and hence we cannot be sure if t = |r - s| Combining the two together - we can definitely say that t = |r-s| and that r-s is positive. Hope this helps

Manat · Answer

@Gladiator59:- Crystal clear. Thank you.

dcwanderer30 · Answer

What's the source for this question?

Gmatprep550 · Answer

Hi  dcwanderer30 ,

Apologies as I don't know the source of above question as someone posted it in whatsapp study group and it was a nice question hence shared with this forum.

lstudentd · Answer

IanStewart  Can you help with if this question were to be written differently in two alternative ways as follows?

Alternative question 1:
Is t = |r-s| ?
1) 3r > 3s
2) t = - (r-s)

Alternative question 2:
Is t = |r-s| ?
1) 3r < 3s
2) t = - (r-s)

IanStewart · Answer

If you just think about two numbers, like 3 and 7, then when we subtract these two numbers in either order, we get either -4, the negative difference of 3 and 7, or we get 4, the positive difference of 3 and 7. If we take the absolute value after we subtract, though, we always get the positive difference: |3 - 7| = |7 - 3| = 4. That's also the distance between 3 and 7 on the number line. Similarly, when we have two different unknowns r and s, then r-s might be the positive difference of r and s, or it might be the negative difference of r and s. But when we take the absolute value, |r - s| is always the positive difference of r and s. So when a question asks "Is t = |r - s|?", the question is asking "Is t equal to the positive difference of r and s?".

I'll simplify your two Alternative Questions in the obvious ways:

Qn 1:
Is t = |r - s|?
1. r - s > 0
2. t = s- r

Here, Statement 2 tells us "t is either the positive or negative difference of r and s". We don't know which. So we can't answer the question. Using both Statements, we know r-s is the positive difference of r and s. So Statement 2 tells us t is equal to the negative difference or r and s, and if t is negative, there's no chance it equals an absolute value of any kind, let alone the positive difference of r and s, so the answer here is C; using both Statements, we can be certain the answer to the question is 'no'.

Qn 2:
Is t = |r - s|?
1. s - r > 0
2. t = s - r

Similarly, Statement 2 tells us "t is either the positive or negative difference of r and s". That's not sufficient alone, but Statement 1 ensures s-r (not r-s) is the positive difference of s and r, so using both Statements, the answer to the original question must be "yes", and the answer is C. Algebraically, if we know Statement 1 is true, we know s - r = |s - r| = |r - s|, so if t = s - r, then t = |r - s|.

If you wanted to test numbers instead, you'd just need to experiment with two scenarios: one where s > r, and one where r > s, and you'd also find C is the answer to each question.

lstudentd · Answer

IanStewart

I have a related question regarding this type of inequality problem. What if we're asked if (y+5)(y-3) > 0? and given the following statements

1) y > 3
2) y < -5

Since y > -5 and y > 3 or y < -5 and y < 3, would neither statement be sufficient?

IanStewart · Answer

The product (y+5)(y-3) will be positive in one of two cases:

- when both factors are positive. Notice that y-3 is always less than y+5 (it's exactly 8 less, no matter what y is), so as long as y-3 is positive, y+5 always will be. So the product will be positive if y - 3 > 0, or if y > 3

- when both factors are negative. Similarly, if y+5 is negative, y-3 will certainly be negative, because it's smaller than y+5. So both factors will be negative if y < -5.

So each statement is sufficient alone in your question, and the answer is D.

edit: and I should have added that the statements are contradictory, as Bunuel points out below, so this is not a possible GMAT question, though you could see either statement alone in a question like this (along with another consistent statement).

Bunuel · Answer

I'd just like to add one thing: this is a flawed question by GMAT's standards. On the GMAT two statements do not contradict each other, while y > 3 and y < -5 clearly do.

lstudentd · Answer

Got it, thanks for pointing out the flaw! What if the question was (y+5)(y-3) < 0 instead and given the following statements: 1) y > -5 2) y > 3 I'm thinking there are two scenarios: A. -5 < y < 3 and B. y < -5 and y > 3. However scenario B is not possible. Would we be able to determine sufficiency? IanStewart Bunuel any chance either of you could weigh in on this?

lstudentd · Answer

IanStewart Was hoping to follow up to see if you could help with a modified question since the previous one was flawed. Thank you so much!

What if the question was (y+5)(y-3) < 0 instead and given the following statements:

1) y > -5
2) y > 3

I'm thinking there are two scenarios: A. -5 < y < 3 and B. y < -5 and y > 3. However scenario B is not possible.

Would we be able to determine sufficiency?

IanStewart · Answer

Your scenario analysis is correct; the inequality (y+5)(y-3) < 0 is true only when -5 < y < 3. So if a DS question asks "is (y+5)(y-3) < 0?", we can rephrase that question more simply as "Is -5 < y < 3?" Statement 1 then is not sufficient; while Statement 1 guarantees that y > -5, we have no way to know whether y < 3 (y might be zero, or might be one million, so the answer can be yes or no). Statement 2, however, is sufficient, because if y > 3 is true, we know with certainty that y is not between -5 and 3, so we can be certain the answer to the question is 'no'.

lstudentd · Answer

Got it, thank you so much  IanStewart !

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Is t =|r - s|? (1) 3r > 3s (2) t = r -s

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