AbdurRakib
If q, s, and t are all different numbers, is q < s < t ?
(1) t - q = |t - s| + |s - q|
(2) t > q
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