Data Sufficiency - Inequalities

1. A. The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone
2. B. The Question can be answered by using either Statement alone
3. C. The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
4. D. The Question cannot be answered even by using both the Statements together

Answer: D

Explanation

We need to determine if P > Q.

**Statement I: P + Q = R + S.**
This equation alone does not allow us to compare P and Q. For example:
– If P=10, Q=5, R=8, S=7 (P+Q=15, R+S=15). Here P > Q.
– If P=5, Q=10, R=7, S=8 (P+Q=15, R+S=15). Here P Q + R.**
This inequality alone does not allow us to compare P and Q. For example:
– If P=10, S=5, Q=8, R=2 (P+S=15, Q+R=10, so 15>10). Here P > Q.
– If P=8, S=5, Q=10, R=2 (P+S=13, Q+R=12, so 13>12). Here P R = P + Q – S
2. P + S > Q + R
Substitute R from (1) into (2):
P + S > Q + (P + Q – S)
P + S > P + 2Q – S
Subtract P from both sides:
S > 2Q – S
Add S to both sides:
2S > 2Q
S > Q.
This combined information tells us that S scored more marks than Q. However, it does not provide any definitive comparison between P and Q. Therefore, even both statements together are insufficient to answer whether P scored more marks than Q. This question tests data sufficiency and algebraic manipulation of inequalities.

1. A. The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone.
2. B. The Question can be answered by using either Statement alone.
3. C. The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone.
4. D. The Question cannot be answered even by using both the Statements together.

Answer: D

Explanation

The question asks if p > q.
Statement 1: p × q > 0. This implies p and q have the same sign (both positive or both negative). If p=3, q=2, then p>q. If p=-2, q=-3, then p>q. But if p=2, q=3, then p<q. If p=-3, q=-2, then p q^2. This implies |p| > |q|. If p=3, q=2, then p>q. If p=-3, q=2, then pq. If p=-3, q=-2, then p |q|. If p and q are both positive, then p > q (e.g., p=3, q=2). If p and q are both negative, then p q. Therefore, the question cannot be answered even by using both statements together.

1. A. The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone.
2. B. The Question can be answered by using either Statement alone.
3. C. The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone.
4. D. The Question cannot be answered even by using both the Statements together.

Answer: C

Explanation

The question asks: Is (p + q – r) > (p – q + r)?
Simplifying this inequality: p + q – r > p – q + r => 2q > 2r => q > r. So, the question is equivalent to asking: Is q > r?
Statement 1: (p – q) is positive, which means p > q. This statement gives no information about the relationship between q and r. Thus, Statement 1 alone is not sufficient.
Statement 2: (p – r) is negative, which means p q. From Statement 2, p < r. Together, these imply q < p < r. From this combined information, we can definitively conclude that q r?’, and we found q < r, the answer is a definitive 'No'. Therefore, both statements together are necessary and sufficient to answer the question.