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

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If q, s, and t are all different numbers, is q < s < t ? [#permalink]

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19 Jun 2017, 04:44
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If q, s, and t are all different numbers, is q < s < t ?

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

(2) t > q
[Reveal] Spoiler: OA

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Sentence Correction-Collection of Ron Purewal's "elliptical construction/analogies" for SC Challenges

Last edited by Bunuel on 19 Jun 2017, 04:59, edited 1 time in total.
Renamed the topic and edited the question.

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Re: If q, s, and t are all different numbers, is q < s < t ? [#permalink]

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19 Jun 2017, 05:28
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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.

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Re: If q, s, and t are all different numbers, is q < s < t ? [#permalink]

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19 Jun 2017, 13:27
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AbdurRakib wrote:
If q, s, and t are all different numbers, is q < s < t ?

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

(2) t > q

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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Kudos [?]: 208 [5] , given: 46

Re: If q, s, and t are all different numbers, is q < s < t ?   [#permalink] 19 Jun 2017, 13:27
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# If q, s, and t are all different numbers, is q < s < t ?

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