Study the following information carefully and answer the given questions :
The number of cricket players in four different academies A, B, C and Dis 120, 150, 100 and 200 respectively.
There are only three types of players in a team: specialist Batsman, specialist bowlers and all-rounders (those who can both bat and ball)
Out of total number of players in academies A, B, C and D, percentage of players who can bat is `75%, 80%, 60% `and `35%` respectively.
Out of total number of players in academies A, B, C and D, percentage of players who can bowl is `50%, 60%, 70%` and `80%` respectively
Fees paid to one specialist batsman for one match is 3times the fees paid to one all-rounder for one match
Fees paid to one specialist bowler for one match is Rs20000.
Respective ratio between the fees paid to one specialist bowler for one match and the fees paid to one all-rounder for one match is 2:3.
What is the respectively ratio between the total fees paid to all the specialist bowlers of academy C and paid to all the all-rounders of academy A?

Options

A. `9:11`
B. `5:7`
C. `11:13`
D. `8:9`

Answer

To solve the problem step by step, we will first calculate the number of batsmen, bowlers, and all-rounders in each academy, and then find the respective fees to answer the question.

### Step 1: Calculate the number of batsmen in each academy
- **Academy A**:
- Total players = 120
- Percentage of batsmen = 75%
- Number of batsmen = \( 120 \times \frac{75}{100} = 90 \)

- **Academy B**:
- Total players = 150
- Percentage of batsmen = 80%
- Number of batsmen = \( 150 \times \frac{80}{100} = 120 \)

- **Academy C**:
- Total players = 100
- Percentage of batsmen = 60%
- Number of batsmen = \( 100 \times \frac{60}{100} = 60 \)

- **Academy D**:
- Total players = 200
- Percentage of batsmen = 35%
- Number of batsmen = \( 200 \times \frac{35}{100} = 70 \)

### Step 2: Calculate the number of bowlers in each academy
- **Academy A**:
- Percentage of bowlers = 50%
- Number of bowlers = \( 120 \times \frac{50}{100} = 60 \)

- **Academy B**:
- Percentage of bowlers = 60%
- Number of bowlers = \( 150 \times \frac{60}{100} = 90 \)

- **Academy C**:
- Percentage of bowlers = 70%
- Number of bowlers = \( 100 \times \frac{70}{100} = 70 \)

- **Academy D**:
- Percentage of bowlers = 80%
- Number of bowlers = \( 200 \times \frac{80}{100} = 160 \)

### Step 3: Calculate the number of all-rounders in each academy
- **Academy A**:
- Total players = 120
- Batsmen + Bowlers = \( 90 + 60 = 150 \)
- All-rounders = \( 120 - 150 = -30 \) (not possible, hence all-rounders = 30)

- **Academy B**:
- Total players = 150
- Batsmen + Bowlers = \( 120 + 90 = 210 \)
- All-rounders = \( 150 - 210 = -60 \) (not possible, hence all-rounders = 60)

- **Academy C**:
- Total players = 100
- Batsmen + Bowlers = \( 60 + 70 = 130 \)
- All-rounders = \( 100 - 130 = -30 \) (not possible, hence all-rounders = 30)

- **Academy D**:
- Total players = 200
- Batsmen + Bowlers = \( 70 + 160 = 230 \)
- All-rounders = \( 200 - 230 = -30 \) (not possible, hence all-rounders = 30)

### Step 4: Calculate the fees for each type of player
- **Fees for Specialist Bowler**: Rs 20,000
- **Fees for All-Rounder**:
- Ratio of fees (Bowler: All-Rounder) = 2:3
- If Bowler = Rs 20,000, then All-Rounder = \( \frac{3}{2} \times 20,000 = 30,000 \)
- **Fees for Specialist Batsman**:
- If All-Rounder = Rs 30,000, then Batsman = \( 3 \times 30,000 = 90,000 \)

### Step 5: Calculate total fees for Specialist Bowlers in Academy C
- Number of Specialist Bowlers in Academy C = 70 (total bowlers) - 30 (all-rounders) = 40
- Total Fees for Specialist Bowlers in Academy C = \( 40 \times 20,000 = 800,000 \)

### Step 6: Calculate total fees for All-Rounders in Academy A
- Number of All-Rounders in Academy A = 30
- Total Fees for All-Rounders in Academy A = \( 30 \times 30,000 = 900,000 \)

### Step 7: Find the ratio of total fees
- Ratio of total fees (Bowlers in C : All-Rounders in A) = \( 800,000 : 900,000 \)
- Simplifying the ratio: \( 8 : 9 \)

### Final Answer
The respective ratio between the total fees paid to all the specialist bowlers of academy C and paid to all the all-rounders of academy A is **8:9**.