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UPSC Prelims 2024CSATLogical ReasoningData Sufficiency - Inequalities

Q70. A Question is given followed by two Statements I and
II. Consider the Question and the Statements. P, Q, R and S appeared in a test. Question : Has P scored more marks than Q ? Statement-I : The sum of the marks scored by P and Q is equal to the sum of the marks scored by R and S. Statement-II : The sum of the marks scored by P and S is more than the sum of the marks scored by Q and R.

Which one of the following is correct in respect of the above Question and the Statements?

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

Detailed Solution

✓ Correct Answer: Option D

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. Since both scenarios are possible, Statement I alone is insufficient. **Statement II: P + S > 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 < Q. Since both scenarios are possible, Statement II alone is insufficient.

**Combining Statement I and Statement II:** 1. P + Q = R + S => 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.

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