Correct Option (3)
The core question requires determining if (p+q−r) is greater than (p−q+r). This inequality can be simplified as follows:
- (p+q−r)>(p−q+r)
- (p+q−r)−(p−q+r)>0
- p+q−r−p+q−r>0
- 2q−2r>0
- 2(q−r)>0
- q−r>0
- q>r
Therefore, the original question is equivalent to asking: Is q>r?
Now, let's evaluate the sufficiency of each statement:
- Statement-1: (p−q) is positive. This implies p−q>0, which simplifies to p>q. This statement provides information about the relationship between p and q, but offers no direct insight into the relationship between q and r. Consequently, Statement-1 alone is not sufficient to answer whether q>r.
- Statement-2: (p−r) is negative. This implies p−r<0, which simplifies to p<r. This statement provides information about the relationship between p and r, but offers no direct insight into the relationship between q and r. Consequently, Statement-2 alone is not sufficient to answer whether q>r.
- Using both Statements together: From Statement-1, we have p>q. From Statement-2, we have p<r. Combining these two inequalities, we establish the relationship q<p<r. This combined information definitively indicates that q<r. Since we have concluded that q<r, we can definitively answer the original question "Is q>r?" with "No". Therefore, both statements together are sufficient to answer the question.
Based on this analysis, the question can be answered by using both statements together, but cannot be answered using either statement alone.
Incorrect Options:
Options (1), (2), and (4) are incorrect. The analysis demonstrates that neither Statement-1 alone nor Statement-2 alone provides sufficient information to determine the relationship between q and r. However, when both statements are combined, a conclusive determination can be made, thereby answering the question definitively.