In a game of 10 players lasting for 30 minutes, five reserves substitute each player, so that all players, including reserves, are on the pitch for the same length of time. How long is each player on the pitch?
The correct answer and explanation is :
To solve this problem, let’s break it down step by step.
Problem Overview:
- There are 10 players and 5 reserves.
- The game lasts for 30 minutes.
- Every player, including reserves, must be on the pitch for the same amount of time.
- The 5 reserves substitute in for the 10 players during the game.
Step 1: Total Playing Time
The total playing time of all players (including reserves) is determined by the total duration of the game (30 minutes) multiplied by the number of players (10 players + 5 reserves = 15 players total). So, the total playing time for all players is:
$$
\text{Total Playing Time} = 30 \, \text{minutes} \times 15 \, \text{players} = 450 \, \text{minutes}
$$
Step 2: Equal Distribution of Time
Since all players, including reserves, are supposed to play the same amount of time, we need to divide the total playing time by the number of players (15) to determine how long each player is on the pitch:
$$
\text{Time Each Player Plays} = \frac{\text{Total Playing Time}}{\text{Number of Players}} = \frac{450 \, \text{minutes}}{15 \, \text{players}} = 30 \, \text{minutes}
$$
Step 3: Understanding Substitutions
The key here is that the reserves substitute each of the 10 players for exactly the same amount of time. This means that even though 5 reserves are rotating in and out, each player (original 10 players and reserves) ends up playing for 30 minutes in total.
Conclusion:
Each player (including the reserves) plays for 30 minutes during the game. The substitutions are arranged so that no player exceeds or falls short of this playing time, ensuring an equal distribution for everyone involved.