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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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.

Could someone please explain the little struggle I'm having with Option B)?

If we want to prove that q < s < t, that means that also q < t (regardless of the placement of s).

Now B tells us, that q > t, hence q < t can't be true at the same time?
Can't I conclude that q < s < t doesn't hold when q > t and clearly answer the question with a no?

What mistake am I making here?

We want to answer whether q < s < t and (2) says that t > q, NOT q > t.
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Oh gosh, how stupid of me... Guess I need a break for the day

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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Bunnuel I tried it in this way . Is it right or wrong

given If q, s, and t are all different numbers, is q < s < t ?
or is 0<s-q<t-q?
Since i am doing the same operation on all the parts on inequality it doesn't violates any rule i suppose

ST 1 t - q = |t - s| + |s - q|
Here t-q is positive ( sum of two mod's) and also greater than s-q as something is been added to |s - q|

Now this equation can be solved in two ways only either same sign or opposite sign of mods
same sign leads t-q= t-s +s-q lhs=rhs therefore 0<s-q<t-q

different signs lead to t-q= t-s -s+q ,,,,,, not possible

or t-q = -t+s+s-q ,,,,,, again lhs is not equal to rhs . this means that the case s-q<0<t-q doesnt holds

Hence st 1 is sufficient
st 2 clearly not suficient hence a

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.

Would anyone please explain this question by plugging in value ?

Thanks in advance.
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Hi,

I have a quick question: Why is (i) is sufficient? Considering we have a total of four mutually exclusive cases with (i) and only one of them follows or is true and rest are not. This fact alone makes (i) insufficient? Please help. Thanks

Bunuel can you pls explain how from this \(t - q = t - 2s + q\) you deduce that \(q=s\)

\(t - q = t - 2s + q\);

Cancel t: \(- q = - 2s + q\);

Re-arrange: \(2s= q + q\);

\(2s=2q\);

\(q=s\).
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You got the answer right, but your approach wouldn't qualify statement 1. Am I wrong to think that way?

1.
one way:
q= 1, s= 2, and t= 3

t-q = lt - sl + ls - ql

3-1 = l3 - 2l + l2 - 1l
2 = l1l + l1l sufficient

second way:
q= 1, s = 3, and t = 2

t-q = lt - sl + ls - ql

2-1 = l2-3l + l3-1l

1 = l1l + l2l

1 does not equal 3, therefore choosing q=1, s=3, and t=2 cannot be right numbers to verify. I get it that t has to be great than s since second way doesn't yield the equation right. But how do we ensure that this will always hold true for any numbers we use?

This is awesome. Made it so much easier to understand. Thank you.

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
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st1: t-q>0 ; t>Q Only one case left. Suff.
st2: t>q ; Nothing given about s. Insuff.

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?