Question

In a football tournament, a player has played a certain number of matches and 10 more matches are to be played. If he scores a total of one goal over the next 10 matches, his overall average will be 0.15 goals per match. On the other hand, if he scores a total of two goals over the next 10 matches, his overall average will be 0.2 goals per match. The number of matches he has played is

Answer 1

Let the number of matches already played be represented by x. In the next 10 matches:

• If one goal is scored in total, the average goals per match becomes 0.15.
• If two goals are scored, the average becomes 0.2.

Let the total number of goals scored in the first \( x \) matches be \( G \).

After playing 10 more matches, the total number of matches becomes: 
\[ x + 10 \] 

Case 1 – Scoring 1 goal in next 10 matches:

Total goals after this case: \( G + 1 \) 
Average goals per match: \( \frac{G + 1}{x + 10} = 0.15 \)

Case 2 – Scoring 2 goals in next 10 matches:

Total goals: \( G + 2 \) 
Average: \( \frac{G + 2}{x + 10} = 0.2 \)

Subtracting these equations:

\[ \frac{G + 2}{x + 10} - \frac{G + 1}{x + 10} = 0.2 - 0.15 \] \[ \frac{(G + 2) - (G + 1)}{x + 10} = 0.05 \] \[ \frac{1}{x + 10} = 0.05 \] 

Solving for \( x \):

\[ x + 10 = \frac{1}{0.05} = 20 \Rightarrow x = 10 \] 

Final Answer:

The number of matches already played is: 10.