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 difference between no. of players who can play tennis & players who play only cricket?

Options

A. 74
B. 64
C. 68
D. None of these

Answer

To solve the problem step by step, we will analyze the information provided and use a Venn diagram to organize the data.

### Step 1: Determine the number of players playing all three games
The total number of players who play tennis is given as 160. According to the problem, 10% of these players play all three games (tennis, cricket, and chess).

**Calculation:**
- Players who play all three games = 10% of 160
- = 0.10 × 160 = 16

### Step 2: Set up the Venn diagram
We will create a Venn diagram with three circles representing tennis, cricket, and chess. We will place the number of players who play all three games (16) in the intersection of all three circles.

### Step 3: Analyze the ratio of cricket to chess players
The ratio of cricket to chess players is given as 3:5. Let’s denote the number of cricket players as 3x and the number of chess players as 5x.

### Step 4: Total of cricket and chess players
It is stated that the total number of cricket and chess players is 100% more than the number of tennis players. Since the number of tennis players is 160, 100% more means:

**Calculation:**
- Total cricket and chess players = 160 + 160 = 320

From the ratio we set up:
- 3x + 5x = 320
- 8x = 320
- x = 40

Thus, the number of cricket players (3x) is:
- Cricket players = 3 × 40 = 120

And the number of chess players (5x) is:
- Chess players = 5 × 40 = 200

### Step 5: Players who play both tennis and chess
It is given that players who play both tennis and chess are 12.5% of total tennis players.

**Calculation:**
- Players who play both tennis and chess = 12.5% of 160
- = 0.125 × 160 = 20

### Step 6: Players who play both tennis and cricket and chess and cricket
The ratio of players who play both tennis & cricket to players who play both chess & cricket is given as 2:3. Let’s denote:
- Players who play both tennis & cricket = 2y
- Players who play both chess & cricket = 3y

### Step 7: Total of players who play both tennis & cricket and chess & cricket
It is also stated that the total of players who play both tennis & cricket and players who play both chess & cricket is equal to one-fourth of chess players.

**Calculation:**
- Total of both = 2y + 3y = 5y
- One-fourth of chess players = 1/4 × 200 = 50

Thus, we have:
- 5y = 50
- y = 10

So, the number of players who play both tennis & cricket is:
- Players who play both tennis & cricket = 2y = 2 × 10 = 20

And the number of players who play both chess & cricket is:
- Players who play both chess & cricket = 3y = 3 × 10 = 30

### Step 8: Calculate the number of players who play only cricket
To find the number of players who play only cricket, we need to subtract those who play both cricket and tennis and those who play both cricket and chess from the total number of cricket players.

**Calculation:**
- Total cricket players = 120
- Players who play both cricket & tennis = 20
- Players who play both chess & cricket = 30

So, players who play only cricket = Total cricket players - (Players who play both cricket & tennis + Players who play both chess & cricket + Players who play all three games)
- = 120 - (20 + 30 + 16)
- = 120 - 66
- = 54

### Step 9: Calculate the difference between players who can play tennis and players who play only cricket
The total number of players who can play tennis is 160. We have already calculated that the number of players who play only cricket is 54.

**Calculation:**
- Difference = Players who play tennis - Players who play only cricket
- = 160 - 54
- = 106

### Final Answer
The difference between the number of players who can play tennis and the players who play only cricket is **106**.

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