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?

Question

Prepp Team · Accepted Answer

Let P, Q, R, and S represent the marks scored by the four individuals P, Q, R, and S in the test.

The question asks: Has P scored more marks than Q? This can be written as: Is \(P > Q\)?

We are given two statements:

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. Mathematically, this is \(P + Q = R + S\).
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. Mathematically, this is \(P + S > Q + R\).

We need to determine if the question \(P > Q\) can be answered using Statement-I alone, Statement-II alone, or both statements together.

Analysis of Statement-I Alone

Statement-I says \(P + Q = R + S\). Let's see if this statement alone is sufficient to answer whether \(P > Q\).

We can rearrange Statement-I as \(P - R = S - Q\). This equation relates the differences in marks, but it doesn't directly tell us the relationship between P and Q.

Consider the following examples that satisfy Statement-I:

Example 1: Let P = 10, Q = 5, R = 8, S = 7. \(P + Q = 10 + 5 = 15\). \(R + S = 8 + 7 = 15\). So, \(P + Q = R + S\) is satisfied. In this case, \(P = 10\) and \(Q = 5\), so \(P > Q\).
Example 2: Let P = 5, Q = 10, R = 7, S = 8. \(P + Q = 5 + 10 = 15\). \(R + S = 7 + 8 = 15\). So, \(P + Q = R + S\) is satisfied. In this case, \(P = 5\) and \(Q = 10\), so \(P < Q\).

Since Statement-I is satisfied in both Example 1 (where \(P > Q\)) and Example 2 (where \(P < Q\)), Statement-I alone is not sufficient to answer the question "Has P scored more marks than Q?".

Analysis of Statement-II Alone

Statement-II says \(P + S > Q + R\). Let's see if this statement alone is sufficient to answer whether \(P > Q\).

We can rearrange Statement-II as \(P - Q > R - S\). This inequality relates differences, but it doesn't directly tell us the relationship between P and Q.

Consider the following examples that satisfy Statement-II:

Example 3: Let P = 10, Q = 4, R = 6, S = 5. \(P + S = 10 + 5 = 15\). \(Q + R = 4 + 6 = 10\). So, \(P + S > Q + R\) (15 > 10) is satisfied. In this case, \(P = 10\) and \(Q = 4\), so \(P > Q\).
Example 4: Let P = 5, Q = 6, R = 7, S = 10. \(P + S = 5 + 10 = 15\). \(Q + R = 6 + 7 = 13\). So, \(P + S > Q + R\) (15 > 13) is satisfied. In this case, \(P = 5\) and \(Q = 6\), so \(P < Q\).

Since Statement-II is satisfied in both Example 3 (where \(P > Q\)) and Example 4 (where \(P < Q\)), Statement-II alone is not sufficient to answer the question "Has P scored more marks than Q?".

Analysis of Both Statements Together

Now, let's use both statements together:

\(P + Q = R + S\)
\(P + S > Q + R\)

From statement (1), we have \(R = P + Q - S\).

Substitute this expression for R into statement (2):

\(P + S > Q + (P + Q - S)\)

\(P + S > Q + P + Q - S\)

Subtract P from both sides:

\(S > Q + Q - S\)

\(S > 2Q - S\)

Add S to both sides:

\(2S > 2Q\)

Divide by 2:

\(S > Q\)

So, using both statements together, we can deduce that S scored more marks than Q (\(S > Q\)). Let's see if we can deduce anything else.

Rearrange statement (1) as \(P - R = S - Q\). Rearrange statement (2) as \(P - Q > R - S\). Since \(S - Q = P - R\), we can write \(R - S = -(P - R) = R - P\). Substituting this into statement (2):

\(P - Q > R - P\)

Add P to both sides:

\(2P - Q > R\)

This inequality \(2P - Q > R\) must hold if both statements are true.

Let's try adding the two statements:

Statement 1: \(P + Q = R + S\)

Statement 2: \(P + S > Q + R\)

Add the left sides and the right sides:

\((P + Q) + (P + S) > (R + S) + (Q + R)\)

\(2P + Q + S > Q + S + 2R\)

Subtract \(Q + S\) from both sides:

\(2P > 2R\)

Divide by 2:

\(P > R\)

So, combining both statements tells us that P scored more marks than R (\(P > R\)) and S scored more marks than Q (\(S > Q\)). Does knowing \(P > R\) and \(S > Q\) allow us to conclude whether \(P > Q\)? Not necessarily.

Let's revisit our examples that satisfied both statements:

Example 1: P=10, Q=5, R=8, S=7. Statement I: \(10 + 5 = 8 + 7\) (15 = 15, True) Statement II: \(10 + 7 > 5 + 8\) (17 > 13, True) Both statements are satisfied. In this case, \(P = 10\) and \(Q = 5\), so \(P > Q\). Also, \(P > R\) (10 > 8) and \(S > Q\) (7 > 5) hold.
Example 2: P=6, Q=8, R=4, S=10. Statement I: \(6 + 8 = 4 + 10\) (14 = 14, True) Statement II: \(6 + 10 > 8 + 4\) (16 > 12, True) Both statements are satisfied. In this case, \(P = 6\) and \(Q = 8\), so \(P < Q\). Also, \(P > R\) (6 > 4) and \(S > Q\) (10 > 8) hold.

Since we can find scenarios that satisfy both Statement-I and Statement-II where \(P > Q\) (Example 1) and scenarios where \(P < Q\) (Example 2), even using both statements together does not provide a definite answer to the question "Has P scored more marks than Q?".

Conclusion on Data Sufficiency

Based on the analysis:

Statement-I alone is not sufficient.
Statement-II alone is not sufficient.
Both Statement-I and Statement-II together are not sufficient.

Therefore, the question cannot be answered even by using both the Statements together.

Statement(s) Used	Sufficient to Answer \(P > Q\)?	Reason
Statement-I alone	No	\(P + Q = R + S\) allows cases where \(P > Q\) and cases where \(P < Q\).
Statement-II alone	No	\(P + S > Q + R\) allows cases where \(P > Q\) and cases where \(P < Q\).
Both Statements together	No	Even when \(P + Q = R + S\) and \(P + S > Q + R\) are true, both \(P > Q\) and \(P < Q\) are possible. (Examples 1 and 2 demonstrate this).

Revision Table: Data Sufficiency Concepts

Concept	Explanation	Sufficiency Check
Data Sufficiency Question	A question followed by statements. The goal is to determine which statement(s) are necessary/sufficient to answer the question.	Is the answer always Yes or always No based on the statement(s)?
Statement Alone is Sufficient	The statement by itself provides enough information to give a unique answer (Yes or No) to the question.	If Statement I alone is sufficient, we don't need to check Statement II or both together.
Both Statements Together are Sufficient	Neither statement alone is sufficient, but when combined, they provide enough information to give a unique answer to the question.	Check if the combination eliminates possibilities that made individual statements insufficient.
Even Both Statements Together are Not Sufficient	Even when combining all given statements, there are still multiple possible answers (Yes or No) to the question.	Demonstrate cases satisfying all statements where the answer to the question differs.

Additional Information: Comparing Quantities

When solving problems that involve comparing quantities based on given equations or inequalities, it's often helpful to:

Translate the question and statements into mathematical expressions (equations or inequalities).
Try to manipulate the expressions to directly compare the quantities in question.
If direct comparison is not straightforward, test specific numerical examples that satisfy the given conditions.
For data sufficiency, to prove insufficiency, you need to find at least two scenarios that satisfy the condition(s) but give different answers to the question (one "Yes" and one "No").
Algebraic manipulation (substitution, addition/subtraction of inequalities) can reveal hidden relationships between variables, as we saw when deducing \(P > R\) and \(S > Q\) from the two statements in this problem. However, these derived relationships might not always be the ones needed to answer the specific question asked.