On the number line shown, is zero halfway

[utopian_wanderer]

On the number line shown, is zero halfway between r and s


<-----------------r---------s----t----------->

1)s is to the right of zero
2)The distance between t and r is the same as the distance between t and -s

In my opinion the answer to this question should be B. However GMAC says that the answer is C.

My reasoning is that, for the distance between rand t to be equal to the distance between t and -s, r should be equal to -s. Therefore statement 2 alone is sufficient to answer this question.

Can someone please explain how the answer is C

[Ian Stewart]

On the number line shown, is zero halfway between r and s


<-----------------r---------s----t----------->

1)s is to the right of zero
2)The distance between t and r is the same as the distance between t and -s

In my opinion the answer to this question should be B. However GMAC says that the answer is C.

My reasoning is that, for the distance between rand t to be equal to the distance between t and -s, r should be equal to -s. Therefore statement 2 alone is sufficient to answer this question.

Can someone please explain how the answer is C

r could be equal to -s, but it doesn't need to be, at least not if all three of our unknowns are negative. We could have r = -4, s = -2 and t = -1, for example.

[sanju09]

On the number line shown, is zero halfway between r and s


<-----------------r---------s----t----------->

1)s is to the right of zero
2)The distance between t and r is the same as the distance between t and -s

In my opinion the answer to this question should be B. However GMAC says that the answer is C.

My reasoning is that, for the distance between rand t to be equal to the distance between t and -s, r should be equal to -s. Therefore statement 2 alone is sufficient to answer this question.

Can someone please explain how the answer is C

r could be equal to -s, but it doesn't need to be, at least not if all three of our unknowns are negative. We could have r = -4, s = -2 and t = -1, for example.

Your reasoning is not totally wrong that "for the distance between r and t to be equal to the distance between t and -s, r should be equal to -s." But you are missing one thing here that it will be true only if we know what type of number (+/-) s is, which we get to know only if we take statement 1 into account also. My answer is C

[navalpike]

Although I am always grateful for Ian’s explanation, I doubt that I would have been able to apply this reasoning on test day. Actually the first time I saw the question, I did not think of the possibility of all three (r, s, & t) being negative.

Is there a more certain (maybe algebraic?) option for solving 2)?
For example, I am sure we could write 2) as

2)The distance between t and r is the same as the distance between t and –s

AB. VALUE ( t – r) = AB. VALUE ( t + s)

(Read : t-r between absolute value bars and t+s between absolute value bars)

Is there an algebraic way from here that might lead to ‘Insufficient’?

[pkw209]

This question takes way longer than 2 minutes to figure out.

Any shortcuts?

[jeemat]

Can anyone explain how this can be accomplished in 2 minutes? Or like they asked above, how they would structure their work?

[Ian Stewart]

Can anyone explain how this can be accomplished in 2 minutes? Or like they asked above, how they would structure their work?

I'd suggest beginning by imagining the possible locations of zero on the number line shown. When we consider statement 2 alone, we know that -s is just the reflection of s through zero on the number line. Certainly s could be to the right of zero, and zero could be halfway between r and s:

---r---0---s---t---

so we just need to see if it's possible that s is to the left of zero. Indeed it is possible, provided all of our numbers are negative:

--r--s--t--0--(-s)-----

When I consider this latter possibility, I just imagine values for r, t and -s so that t is in the middle - for example -6, -1, 4, (so s = -4) or -10, -2, 6 (so s = -6). It's certainly feasible to imagine one of these scenarios within two minutes, provided you look at the problem in a suitable way early on.

[Scott@TargetTestPrep]

On the number line shown, is zero halfway between r and s


<-----------------r---------s----t----------->

1)s is to the right of zero
2)The distance between t and r is the same as the distance between t and -s

Solution:

We need to determine whether zero halfway between r and s. We see that 0 could be in one of the following four places: 1) to the left of r, 2) between r and s, 3) between s and t, 4) to the right of t. We will refer to these four cases as cases 1, 2, 3 and 4, respectively.

Statement One Only:

s is to the right of zero.

This tells us s is positive; however, without knowing anything about r, we can’t determine whether they are opposites. Statement one is not sufficient to answer the question.

Statement Two Only:

The distance between t and r is the same as the distance between t and -s.

We see that it can’t be case 1 or 3 since the former case has t further away from -s than it’s from r whereas the latter case has t further away from r than it’s from s. However, it can still be either case 2 or 4. If it’s the former, then yes, not only 0 is between r and s, 0 is exactly halfway between r and s. However, if it’s the latter, then no, because both r and s are less than 0. Statement two is not sufficient to answer the question.

Statements One and Two Together:

From statement two, we know it’s either case 2 or 4. However, since from statement one, s is positive, then it must be case 2 (since this case has s positive) and not case 4 (since this case has s negative). From the analysis for statement two, we see that if it is case 2, then 0 is exactly halfway between r and s. The two statements together are sufficient to answer the question.

Answer: C