Analyzing the Football Match Data Sufficiency Problem

The question asks whether Team P wins a football match against Team Q, given that P was 3 goals behind Q with 10 minutes remaining. We are provided with two statements and need to determine if either statement alone or both statements together are sufficient to answer the question.

Understanding the Initial Situation

With 10 minutes left in the match, Team P was 3 goals behind Team Q. Let's denote the score at this point as follows:

• Score of Team P with 10 minutes left: $S_{P, 10}$
• Score of Team Q with 10 minutes left: $S_{Q, 10}$

According to the question, Team P was behind by 3 goals, so: $S_{Q, 10} - S_{P, 10} = 3$

We want to know if Team P wins the match. This means we need to compare the final score of Team P ($S_{P, final}$) with the final score of Team Q ($S_{Q, final}$). Team P wins if $S_{P, final} > S_{Q, final}$.

Analyzing Statement I Alone

Statement I: Team P Scored 4 goals in the last 10 minutes.

This statement gives us information about Team P's scoring in the final part of the match. P's final score is the score it had with 10 minutes left plus the goals scored in the last 10 minutes:

$S_{P, final} = S_{P, 10} + 4$

However, Statement I gives no information about whether Team Q scored any goals in the last 10 minutes. Let's denote the goals scored by Q in the last 10 minutes as $G_{Q, last 10}$. Q's final score would be:

$S_{Q, final} = S_{Q, 10} + G_{Q, last 10}$

We know $S_{Q, 10} = S_{P, 10} + 3$. Substituting this into the equation for $S_{Q, final}$:

$S_{Q, final} = (S_{P, 10} + 3) + G_{Q, last 10}$

Team P wins if $S_{P, final} > S_{Q, final}$. Substituting the expressions for final scores:

$S_{P, 10} + 4 > (S_{P, 10} + 3) + G_{Q, last 10}$

Simplifying the inequality:

$4 > 3 + G_{Q, last 10}$

$1 > G_{Q, last 10}$

For Team P to win, the number of goals scored by Team Q in the last 10 minutes ($G_{Q, last 10}$) must be less than 1. Since the number of goals must be a whole number, this means $G_{Q, last 10}$ must be 0.

Statement I only tells us about P's scoring. It does not tell us if Q scored 0 goals, 1 goal, or more goals in the last 10 minutes.

• If $G_{Q, last 10} = 0$, then P wins.
• If $G_{Q, last 10} = 1$, then $1 > 1$ is false, so P does not win (it would be a draw as we'll see later).
• If $G_{Q, last 10} > 1$, then P does not win (Q would lead).

Since Statement I alone does not provide information about $G_{Q, last 10}$, we cannot definitively determine if $1 > G_{Q, last 10}$ is true. Therefore, Statement I alone is not sufficient to answer the question.

Analyzing Statement II Alone

Statement II: Team Q Scored a total of 4 goals in the match.

This statement gives us Team Q's final score:

$S_{Q, final} = 4$

We still have the initial information that $S_{Q, 10} - S_{P, 10} = 3$. Statement II tells us $S_{Q, final} = 4$, but it does not give us any information about Team P's scoring throughout the match, especially in the last 10 minutes, which is crucial. We know $S_{P, final} = S_{P, 10} + G_{P, last 10}$, where $G_{P, last 10}$ are the goals P scored in the last 10 minutes.

Knowing $S_{Q, final}$ and the score difference with 10 minutes left does not allow us to determine $S_{P, final}$. We don't know $S_{P, 10}$, $S_{Q, 10}$, or $G_{P, last 10}$.

For example:

• If $S_{Q, 10} = 4$, then $S_{P, 10} = 1$. Q scored 0 in the last 10 minutes ($4+0=4$). If P scored 3 in the last 10 minutes, $S_{P, final} = 1+3=4$. Final score P=4, Q=4. Draw.
• If $S_{Q, 10} = 3$, then $S_{P, 10} = 0$. Q scored 1 in the last 10 minutes ($3+1=4$). If P scored 5 in the last 10 minutes, $S_{P, final} = 0+5=5$. Final score P=5, Q=4. P wins.

Statement II alone does not provide enough information about Team P's final score to compare it with Team Q's final score. Therefore, Statement II alone is not sufficient to answer the question.

Analyzing Both Statements Together

Let's combine the information from both statements:

From the question: $S_{Q, 10} - S_{P, 10} = 3$ From Statement I: $G_{P, last 10} = 4$, so $S_{P, final} = S_{P, 10} + 4$ From Statement II: $S_{Q, final} = 4$

We also know that Q's final score is its score with 10 minutes left plus goals scored in the last 10 minutes:

$S_{Q, final} = S_{Q, 10} + G_{Q, last 10}$

Substitute $S_{Q, final} = 4$ into this equation:

$4 = S_{Q, 10} + G_{Q, last 10}$

Also, from the initial condition, $S_{Q, 10} = S_{P, 10} + 3$. Substitute this into the equation above:

$4 = (S_{P, 10} + 3) + G_{Q, last 10}$

Simplify:

$1 = S_{P, 10} + G_{Q, last 10}$

We want to determine if $S_{P, final} > S_{Q, final}$, which is $S_{P, 10} + 4 > 4$. This simplifies to $S_{P, 10} > 0$.

From the equation $1 = S_{P, 10} + G_{Q, last 10}$, considering that scores must be non-negative whole numbers ($S_{P, 10} \ge 0$ and $G_{Q, last 10} \ge 0$), we have possible pairs for $(S_{P, 10}, G_{Q, last 10})$ that sum to 1:

• Case 1: $S_{P, 10} = 1$ and $G_{Q, last 10} = 0$.
  • If $S_{P, 10} = 1$, then from $S_{Q, 10} - S_{P, 10} = 3$, we get $S_{Q, 10} - 1 = 3$, so $S_{Q, 10} = 4$.
  • Check consistency: $S_{Q, final} = S_{Q, 10} + G_{Q, last 10} = 4 + 0 = 4$. This matches Statement II.
  • In this case, $S_{P, final} = S_{P, 10} + G_{P, last 10} = 1 + 4 = 5$. $S_{Q, final} = 4$. P wins (5 > 4).
• Case 2: $S_{P, 10} = 0$ and $G_{Q, last 10} = 1$.
  • If $S_{P, 10} = 0$, then from $S_{Q, 10} - S_{P, 10} = 3$, we get $S_{Q, 10} - 0 = 3$, so $S_{Q, 10} = 3$.
  • Check consistency: $S_{Q, final} = S_{Q, 10} + G_{Q, last 10} = 3 + 1 = 4$. This matches Statement II.
  • In this case, $S_{P, final} = S_{P, 10} + G_{P, last 10} = 0 + 4 = 4$. $S_{Q, final} = 4$. The match is a draw (4 = 4). P does not win.

We have found two scenarios that are consistent with both statements:

Since in Case 1, P wins, and in Case 2, P does not win (it's a draw), even combining both statements does not allow us to definitively answer whether Team P wins the match. The answer depends on how the goals leading up to the score difference of 3 and Q's total of 4 were distributed over the match duration, specifically how many goals Q scored in the last 10 minutes.

Conclusion on Data Sufficiency

Neither Statement I alone, nor Statement II alone, nor both statements together provide sufficient information to determine if Team P wins the match. The crucial missing piece is information about Team Q's scoring in the last 10 minutes.

Revision Table

Additional Information on Football Match Scores and Data Sufficiency

This problem is an example of a data sufficiency question often found in logical reasoning or quantitative sections of tests. The key is to determine if the given information *uniquely* leads to a single conclusion (either "Yes, P wins" or "No, P does not win"). If multiple outcomes are possible based on the statements, then the information is not sufficient.

In football, scores are always non-negative integers. This constraint is important when considering possible score combinations. A draw occurs when $S_{P, final} = S_{Q, final}$. Losing occurs when $S_{P, final} < S_{Q, final}$. Winning occurs when $S_{P, final} > S_{Q, final}$. The question specifically asks "Does team P win the match?", which requires the strict inequality $S_{P, final} > S_{Q, final}$ to be definitively true.

Score progression matters. A score of (1, 4) with 10 minutes left means P had 1 goal and Q had 4 goals at that specific moment. A total score of 4 for Q means their goals accumulated over the entire match sum to 4. How that total of 4 was reached (e.g., 3 goals by the 80th minute and 1 more goal in the last 10 minutes, or all 4 goals scored before the 80th minute) impacts the score difference at the 80th minute and is relevant when combined with information about scoring in the last 10 minutes.