Study the given passage carefully & answer the questions in a sport Academy XY', there are some student who can play three games i.e tennis, cricket & chess. Total number of players who play tennis is 160 & all three games are played by 10% of total tennis players. Ratio of cricket to chess players is 3:5 and total of cricket & chess players is 100% more than tennis players. Players who play both tennis and chess are `12(1)/(2)%` of total tennis players. Ratio of players who play both tennis & cricket to players who play both chess & cricket is 2:3 & total of players who play both tennis & cricket and players who play both chess & cricket is equal to one-fourth of chess players.
What is the average no. of players who play only one game?

Options

A. `139(2)/(3)`
B. `129(1)/(3)`
C. 135
D. None of these

Answer

To solve the problem step by step, we will break down the information provided and calculate the average number of players who play only one game.

### Step 1: Determine the number of players for each game
- Total players who play tennis = 160
- Players who play all three games (Tennis, Cricket, Chess) = 10% of Tennis players = 10% of 160 = 16 players.

### Step 2: Calculate the total number of players for Cricket and Chess
- The ratio of Cricket to Chess players is given as 3:5.
- Let the number of Cricket players be 3x and Chess players be 5x.
- The total number of Cricket and Chess players is 100% more than Tennis players:
- Total Cricket + Chess = 160 + 160 = 320.

So, we have:
\[ 3x + 5x = 320 \]
\[ 8x = 320 \]
\[ x = 40 \]

Now, we can find the number of players:
- Number of Cricket players = 3x = 3(40) = 120
- Number of Chess players = 5x = 5(40) = 200

### Step 3: Determine the players who play both Tennis and Chess
- Players who play both Tennis and Chess = 12.5% of total Tennis players = 12.5% of 160 = 20 players.

### Step 4: Determine the players who play both Tennis and Cricket and both Chess and Cricket
- Let the number of players who play both Tennis and Cricket be 2y and those who play both Chess and Cricket be 3y (from the ratio 2:3).
- The total of players who play both Tennis & Cricket and Chess & Cricket is equal to one-fourth of Chess players:
- Total = 1/4 of Chess players = 1/4 of 200 = 50.

So, we have:
\[ 2y + 3y = 50 \]
\[ 5y = 50 \]
\[ y = 10 \]

Now we can find the number of players:
- Players who play both Tennis and Cricket = 2y = 2(10) = 20
- Players who play both Chess and Cricket = 3y = 3(10) = 30

### Step 5: Calculate the number of players who play only one game
- Players who play only Tennis = Total Tennis - (Players who play both Tennis & Cricket + Players who play both Tennis & Chess + Players who play all three)
- = 160 - (20 + 20 + 16) = 160 - 56 = 104

- Players who play only Cricket = Total Cricket - (Players who play both Tennis & Cricket + Players who play both Chess & Cricket + Players who play all three)
- = 120 - (20 + 30 + 16) = 120 - 66 = 54

- Players who play only Chess = Total Chess - (Players who play both Chess & Cricket + Players who play both Tennis & Chess + Players who play all three)
- = 200 - (30 + 20 + 16) = 200 - 66 = 134

### Step 6: Calculate the average number of players who play only one game
- Total players who play only one game = Players who play only Tennis + Players who play only Cricket + Players who play only Chess
- = 104 + 54 + 134 = 292

- Average number of players who play only one game = Total players who play only one game / 3
- = 292 / 3 = 97.33

### Final Answer
The average number of players who play only one game is approximately **97.33**.

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