Is zero halfway between r and s

[Guest]

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

THE ANSWER IS C/ GMAT-PREP

I assumed statement 2 alone was sufficient, however I was wrong. My question is what is the best approach or method, in short amount of time, in going about solving this problem? Your input is appreciated....

[Guest]

Additional question: When taken together statement 1 and 2, is it ok to create an equation out of statement 2?

That is, t-r=t-(-s) the two t's cancel out and you are left with -r=s/ s=+ve t=+ve r=-ve
Then I plugged in numbers to determine whether the two statements together is sufficient to answer the question.

Furthermore, I assumed statement 2 alone was sufficient using the equation -r=s, however I was wrong. Perhaps, I missed something. Could please clarify this for me. Your input and assistance is greatly appreciated.

[RonPurewal]

i would always think about these things SPATIALLY / VISUALLY at first, and set up algebraic equations only as a "plan b". the problem with algebraic equations is that it's too easy to fall into traps.

the particular trap you've fallen into in your interpretation of (2) is that of assuming "-s" is to the LEFT of "t". there is no good reason whatsoever to make this assumption, and, what's more, at least one good reason (viz., "the gmat loves to test exactly these sorts of assumptions) not to make it.
of course, you don't need reasons to be very careful about your assumptions; that should be your default state.

if "-s" is to the right of "t", then you have
<--r-------s---t-----------(-s)-->
in which case 0 is in no-man's-land between "t" and "-s".
in this case, note that "s" is negative. also note that (-s) is positive in this case, a situation that is difficult to digest for most students.

taking statements (1) and (2) together eliminates the above possibility, leaving only the case that you have outlined.

--

incidentally, the fault in your algebraic approach lies in writing the distance between t and (-s) as t - (-s). this writing is correct only if t is greater than (-s), an assumption that, as we've seen, is unjustified.
the correct way to write the distance is |t - (-s)| = |t + s|, an expression that is thoroughly unhelpful in solving this problem.

[kramacha1979]

How do you approach such problems ? The strategy guides does not focus too much on these types of questions ?

[RonPurewal]

How do you approach such problems ? The strategy guides does not focus too much on these types of questions ?

the best way to approach these problems, in my opinion, is encapsulated by what i've posted above:

i would always think about these things SPATIALLY / VISUALLY at first, and set up algebraic equations only as a "plan b". the problem with algebraic equations is that it's too easy to fall into traps.

there you go. you should try to develop your spatial understanding of the problem, and should reserve algebraic solutions for problems that have no ready spatial interpretation.

you should also make sure that you consider number properties, heavily, when you approach these problems.
many of these number-line problems hinge fundamentally on whether certain numbers are positive or negative, or, similarly, whether unknowns are placed to the left or right of each other on the number line.

--

incidentally, pretty much exactly the same things are true for COORDINATE problems (i.e., problems in the x-y plane):
* again, you should be able to interpret most problem statements SPATIALLY
* you should reserve algebraic solutions for problems that are clearly intended as algebraic, such as statement 1 of this problem. in that statement, not only is the comparison illogical from a spatial standpoint (you can't intuitively compare a slope to an intercept), but the two quantities in the statement, slope and y-intercept, are EXACTLY the quantities that appear in the standard form of the equation of a line (y = mx + b).
unless a problem is that obviously intended to be algebraic, don't solve it with algebra except as a last resort.

[loganscalder]

I'd like to disagree with Ron Purewal if that is ok. He said, "the correct way to write the distance is |t - (-s)| = |t + s|, an expression that is thoroughly unhelpful in solving this problem." That is the correct way to write the distance between t and s and it IS helpful.

Statement 2 gives |t-r| = |t+s|
Now to solve this absolute value equation we break it into equations with out absolute values:
i) t-r = -(t+s)
and
ii) t-r = t+s
These two equations identify the only times that the absolute value equation will be true.
Rearrange these equations and get:

i)2t+s=r In order for this to be true either s or both s and t would have to be negative (on the left of zero) and therefore zero is not half way between r and s

ii) -r=s If this is the case, then zero is half way between r and s (and s must be on the right of zero).

Now we need something to help us know if s is on the right or left of zero. This is supplied by Statement one.

I think learning the algebra can be a helpful way to solve this problem.

Thoughts?

[ghong14]

i)2t+s=r In order for this to be true either s or both s and t would have to be negative (on the left of zero) and therefore zero is not half way between r and s

I don't see how you could have deducted that from the equation. Care to explain? Thanks

[loganscalder]

i)2t+s=r In order for this to be true either s or both s and t would have to be negative (on the left of zero) and therefore zero is not half way between r and s

I don't see how you could have deducted that from the equation. Care to explain? Thanks

Yeah, sorry.
Here is where 2t+s=r comes from (to be safe, I'll start with the original absolute value equation):
We originally had the absolute value equation: |t-r|=|r+s|. This equation will be true in two cases:
a) t-r = -(t+s)
and
b) t-r= t+s

in case 'a', t-r=-(t+s) => t-r= -t-s => 2t+s=r (by adding t, r, and s to both sides of the equation)

2t+s=r means that both s and t are negative, otherwise, they could not add to a number to their right on the number line.

[ghong14]

I think at this point it is pretty clear why A is not sufficient. So I will just point out two scenarios that make B insufficient.

Let's say that 0 is between R and S then we can use the following scenario

R.....0.......S.......T
-1....0.......1.......2

Here |2-(-1)|=|2-(-(1))| Notice that S equals 1 but -S equals -1
This would be a YES to the question. 0 is halfway between R and S and statement 2 is satisfied.

Let's say that T is to the right of zero

R.....S......T......0
-4....-2.....-1.....0

Here |-1-(-4)|=|-1-(-(-2))| Notice that S equals -2 but -S equals 2. This would be a NO to the question. 0 IS NOT halfway between R and S. Statement 2 insufficient.

Hence B along is not sufficient.

[RonPurewal]

I think at this point it is pretty clear why A is not sufficient. So I will just point out two scenarios that make B insufficient.

Let's say that 0 is between R and S then we can use the following scenario

R.....0.......S.......T
-1....0.......1.......2

Here |2-(-1)|=|2-(-(1))| Notice that S equals 1 but -S equals -1
This would be a YES to the question. 0 is halfway between R and S and statement 2 is satisfied.

Let's say that T is to the right of zero

R.....S......T......0
-4....-2.....-1.....0

Here |-1-(-4)|=|-1-(-(-2))| Notice that S equals -2 but -S equals 2. This would be a NO to the question. 0 IS NOT halfway between R and S. Statement 2 insufficient.

Hence B along is not sufficient.

these are good examples; thanks for posting them.

but... why would you use absolute-value expressions to find the distance between -1 and 2?
that's something you could have found easily in the fourth grade, and, needless to say, you wouldn't have needed absolute values to find it.
let's not make this any harder than it needs to be.

[mamajonov]

Does the statement B mean that -s equal to r?
In addition, is the asnwer to this question "YES" or "NO" in choise C.
Thanks in advance

[RonPurewal]

Does the statement B mean that -s equal to r?
In addition, is the asnwer to this question "YES" or "NO" in choise C.
Thanks in advance

Please read this whole thread. The answers to these questions are already here.

Thanks.