If q, s, and t are all different numbers, is q < s < t ?

With less math:

Start with statement 2. This is insufficient, since s could be greater than t, or s could be between q and t. Eliminate answers B and D.

Statement 1: When you see |x - y|, think 'distance between x and y on the number line'. That's all that means. So, this statement says that t - q is equal to the distance between t and s, plus the distance between s and q. In other words, s has to be between t and q.

Jot down some diagrams on your paper to convince yourself of that: in order for the distances to make sense, s has to be in the middle.



Also, t-q has to be positive, since it's the sum of two absolute values. So, t is greater than q.

If t is greater than q and s is in the middle, you know that q < s < t. Sufficient.
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If q, s, and t are all different numbers, is q < s < t ?

(1) t - q = |t - s| + |s - q|

Notice that the right hand side is positive (it's the sum of two absolute values, so two non-negative values, in fact, in our case two positive values, since we know that the variables are distinct). Thus the left hand side must also be positive, which means that t > q. So, we can have 3 cases for s:

a. ---s---q-------t-------
In this case \(s < q < t\):
\(t - s > 0\) and \(s - q < 0\), which would mean that \(|t - s| = t -s\) and \(|s - q| = -(s - q)\) (recall that |x| = x when x > 0 and x = -x when x <= 0).
So, \(|t - s| + |s - q| = (t -s) - (s - q) = t - 2s + q\).

So, in his case we'd have \(t - q = t - 2s + q\) or \(q=s\). But we are told that q, s, and t are all different numbers, so this case is out.

b. -------q---s---t-------
In this case \(q < s < t\):
\(t - s > 0\) and \(s - q > 0\), which would mean that \(|t - s| = t -s\) and \(|s - q| = s - q\). So, \(|t - s| + |s - q| = (t -s) + (s - q) = t - q\).

This matches the info given in the statement.

c. -------q-------t---s---
In this case \(q < t < s\):
\(t - s < 0\) and \(s - q > 0\), which would mean that \(|t - s| = -(t -s)\) and \(|s - q| = s - q\). So, \(|t - s| + |s - q| = -(t -s) + (s - q) = -t + 2s - q\).

So, in his case we'd have \(t - q = -t + 2s - q\) or \(t=s\). But we are told that q, s, and t are all different numbers, so this case is out.

Only q < s < t case is possible. Sufficient.

(2) t > q. Not sufficient.

Answer: A.
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We need to determine whether q < s < t, given that q, s, and t are all different numbers.

Statement One Alone:

t - q = |t - s| + |s - q|

Since q, s, and t are different numbers, both |t - s| and |s - q| are positive quantities, and their sum |t - s| + |s - q| will also be positive. This also makes the left-hand side t - q positive. Since t - q > 0, we have t > q.

We know t > q, but we still have to determine whether s is between them. That is, is q < s < t? We have three scenarios to consider.

(1) If q < s < t, then t > s and s > q, and then:

t - q = t - s + s - q

t - q = t - q

We see that this equation holds true: t - q = |t - s| + |s - q|, and furthermore q < s < t.

(2) If s < q < t, then t > s and q > s, and thus t -s is positive while s - q is negative, and we have:

|t - s| + |s - q|

t - s + [-(s - q)]

t - s - s + q

t - 2s + q ≠ t - q

Since t - 2s + q ≠ t - q, the equation does not hold and we can’t have s < q < t.

(3) If q < t < s, then s > t and s > q, and thus t - s is negative while s - q is positive, and we have:

|t - s| + |s - q|

-(t - s) + s - q

-t + s + s - q

-t + 2s - q

Since -t + 2s - q ≠ t - q, we see that the equation does not hold, so we can’t have q < t < s.

We see that only scenario 1 is true if t - q = |t - s| + |s - q,| and we do have q < s < t. Statement one alone is sufficient.

Statement Two Alone:

t > q

We know t > q, but we still have to determine whether s is between them. It’s possible that q < s < t, but it is also possible that s < q < t or q < t < s. Since we don’t know anything about s, we can’t determine which case is valid. Statement two alone is not sufficient.

Answer: A
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the fastest way to solve this is to draw the number line
q------------t-----s
look at 1
absolute value is the length of the line section
if s is in between, the answer is yes
s can not be outside qt because if s is so, we can not have condition 1. remember /s-q/= the line of sq.
length sq can not be length of sq+length of ts.

that is all.

This seems pretty easy and faster with picking numbers, try numbers in the given order 1,2,3 assigned for q,s,t respectively then try with a different order 1,3,2. In first case the equation in the first option will satisfy i.e LHS=RHS, while in the second case it will fail.

Hence the the numbers should be in ascending order to satisfy the first option, so 1st option is suff.

In second case no detail about s, so insuff.

So answer is A.

Given: q, s, and t are all different numbers

Target question: Is q < s < t ?

Statement 1: t - q = |t - s| + |s - q|
Since q, s, and t are all different numbers, we know that |t - s| is POSITIVE, and |s - q| is POSITIVE.
So, t - q = some positive number
From this we can conclude that: q < t
On the number line we have something like this:

From here we need only determine whether s is between q and t

To help us we can use a nice property that says: |x - y| = the distance between x and y on the number line
For example: |3 - 10| = 7, so the distance between 3 and 10 on the number line is 7

So, the statement "t - q = |t - s| + |s - q|" tells us that: (the distance between t and q) = (the distance between t and s) + (the distance between s and q)
The ONLY time this equation holds true is when is between q and t

Given this, it MUST be the case that q < s < t

Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: t > q
Since there is no information about s, we cannot answer the target question with certainty.
Statement 2 is NOT SUFFICIENT

Answer: A

Cheers,
Brent

Hello ccooley, can you let me know that for statement 1, why you did not consider a case where s was greater than both q and t?

It's a very simple one (I got this wrong though)

If q, s, and t are all different numbers, is q < s < t ?

(1) t - q = |t - s| + |s - q|

(2) t > q

what we can infer from 1 is that s is between t and q

so it can be

t-----------s-------------q

or

q-----------s------------t

since RHS is positive, LHS too must be poistive,

hence

t-q>0

hence

t>q

thus first option remains..

Option A..

Bingo! No maths needed...

|x-y| = the distance between x and y



Statement 1: t - q = |t - s| + |s - q|
Given that q, s, and t are different numbers and that absolute values cannot be NEGATIVE, the left side of the equation must represent a POSITIVE value:
t-q > 0
t > q

Plotted on number line:
q----------t

Statement 1 requires that |t - s| + |s - q| = the blue distance above.
If s is to the left of q, then |t-s| -- the distance between t and s -- will EXCEED the blue distance above.
If s is to the right of t, then |s-q| -- the distance between s and q -- will EXCEED the blue distance above.
Since s can be neither to the left of q nor to the right of t, s must be BETWEEN q and t, with the result that q < s < t.
Thus, the answer to the question stem is YES.
SUFFICIENT.

Statement 2:
No information about s.
INSUFFICIENT.


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Am i understanding it correctly that if we had t-q in module in first statement it wouldn't be sufficient?

Hi Bunuel

Could you please share similar questions for practice?

Thank you!

10. Absolute Value

• Theory
  Math: Absolute value (Modulus)
  Absolute Value: Tips and hints
  Properties of Absolute Values on the GMAT
  Absolute modulus : A better understanding
  GMAT Question Patterns: Abolute Value
  The Reason Behind Absolute Value Questions on the GMAT
  
  • Questions
    Inequality and absolute value questions from my collection
    The E-GMAT Question Series on ABSOLUTE VALUE
    DS Questions
    PS Questions

For more check Ultimate GMAT Quantitative Megathread

Hope it helps.
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Solution:

St (1)

Given

Distance (t, q) = distance (t, s) + distance (s,q)

t-q shall be positive, as it's the sum of two absolute values.
So, t > q.

t > q and s is at the center, thus q < s < t. Sufficient.

St (2)

It gives no information of s. s>q or s<q. Thus Insufficient.

option (a)



Hope this helps
Devmitra Sen(Quants GMAT Expert)
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Video solution from Quant Reasoning:
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CrackVerbalGMAT In Statement I, how did you conclude that s is in between?

With St(1) we have t - q = |t - s| + |s - q|

We know |t - s| = |s - t|
=>Either t > s or s > t but if s>t then |s - q| > |t - q| which invalidates given condition.

Also, |s - q| = |q - s|
=>Either q > s or s > q but if q>s then |t - s| > |t - q| which invalidates given condition

=> |t-s| = t-s and |s-q| = s-q
=> t>s & s>q
=> t>s>q
=>s is in between

Hope you are clear on this!
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Answering this question since I didn’t see an answer yet.

I think that the author did consider that case, just that he didn’t include it in the image.

Let's look at the case:

Say, s is the largest. t is anyway greater than q since t - q is the sum of two positive values.

So, the numbers would appear like:
q …… t …… s
on the number line.

Now, the distance between q and s (|s - q|) is already more than the distance between t and q (t - q). On top of that we're adding a positive value (|t - s|) to the already larger value. So, if s were the greatest, statement 1 would not be valid. Thus s can't be the greatest.

In DS questions, our job is not to check the validity of the statements. Our job is to take the statements as given information and then check whether we can answer the main question with absolute certainty.

Hope that clarifies.

Do you mean if the first statement read:

|t - q| = |t - s| + |s - q| ?

That's a good question. Let's explore.

So, [the distance between t and s] plus [the distance between s and q] is equal to [the distance between t and q]. So, certainly, s would still be in the middle.

I'm imagining three towns on a straight road: A, B and C. If the distance between towns A and C is equal to the sum of the distance between A and B and distance between B and C, B will certainly be in the middle of A and C. However, which one of A and C is on the right of B and which one on the left of B is not clear.

Which one of t and q is on the right on the number line is no longer evident. Either one of them could be the greatest. So, yes, statement 1 is no longer sufficient.

What do you think the answer would be if statement 1 were changed the way you mentioned?

Bunuel KarishmaB

What would be the answer if given that q, s & t are not all different numbers or q,s, t could be anything (no such condition given)

For st 1, we can still reject case 1:
Consider q<t<s,
t-q=s-t+s-q
t=s, but s>t, so we can reject this case.

Is my solution correct ?