Directions: Given data shows total male and female doctors in three hospital in a seminar. Read data carefully and answer the question:
In annual seminar of three hospitals, A, B and C some male and female doctors represent their hospitals. Average number of female doctors who represent A and B is 210. Total male doctors in A and B is 810. Number of female doctors is `2/3`rd and `2/5`th of male doctor in A and B repectively. Total female doctor who represent C are `25%` more than total female doctor who represent A and total male doctor who represent C are `33 1/3%` more than female doctor who represent B.
Find the ratio between total male doctors who represent B and C together to total female doctors who represent A and C together?

Options

A. `23:13`
B. `23:14`
C. `23:18`
D. `23:12`

Answer

To solve the problem step by step, let's break down the information given and derive the required values systematically.

### Step 1: Define Variables
Let:
- \( M_A \) = Number of male doctors in Hospital A
- \( M_B \) = Number of male doctors in Hospital B
- \( F_A \) = Number of female doctors in Hospital A
- \( F_B \) = Number of female doctors in Hospital B
- \( F_C \) = Number of female doctors in Hospital C
- \( M_C \) = Number of male doctors in Hospital C

### Step 2: Set Up Equations
From the problem statement, we have the following information:

1. Total male doctors in A and B:
\[ M_A + M_B = 810 \]

2. Average number of female doctors in A and B:
\[ \frac{F_A + F_B}{2} = 210 \implies F_A + F_B = 420 \]

3. Female doctors in A are \( \frac{2}{3} \) of male doctors in A:
\[ F_A = \frac{2}{3} M_A \]

4. Female doctors in B are \( \frac{2}{5} \) of male doctors in B:
\[ F_B = \frac{2}{5} M_B \]

### Step 3: Substitute and Solve for \( M_A \) and \( M_B \)
Substituting \( F_A \) and \( F_B \) into the equation \( F_A + F_B = 420 \):
\[ \frac{2}{3} M_A + \frac{2}{5} M_B = 420 \]

To eliminate the fractions, find a common denominator (which is 15):
\[ 10M_A + 6M_B = 6300 \quad \text{(Multiplying through by 15)} \]

Now we have two equations:
1. \( M_A + M_B = 810 \)
2. \( 10M_A + 6M_B = 6300 \)

### Step 4: Solve the System of Equations
From the first equation, express \( M_A \):
\[ M_A = 810 - M_B \]

Substituting into the second equation:
\[ 10(810 - M_B) + 6M_B = 6300 \]
\[ 8100 - 10M_B + 6M_B = 6300 \]
\[ 8100 - 4M_B = 6300 \]
\[ 4M_B = 1800 \implies M_B = 450 \]

Now substitute back to find \( M_A \):
\[ M_A = 810 - 450 = 360 \]

### Step 5: Calculate Female Doctors in A and B
Now calculate \( F_A \) and \( F_B \):
\[ F_A = \frac{2}{3} \times 360 = 240 \]
\[ F_B = \frac{2}{5} \times 450 = 180 \]

### Step 6: Calculate Female Doctors in C
Total female doctors in C are 25% more than total female doctors in A:
\[ F_C = F_A + 0.25F_A = 240 + 60 = 300 \]

### Step 7: Calculate Male Doctors in C
Total male doctors in C are \( 33 \frac{1}{3}\% \) more than female doctors in B:
\[ M_C = F_B + \frac{1}{3}F_B = 180 + 60 = 240 \]

### Step 8: Calculate the Required Ratio
Now we need to find the ratio of total male doctors in B and C together to total female doctors in A and C together:
\[ \text{Total male doctors in B and C} = M_B + M_C = 450 + 240 = 690 \]
\[ \text{Total female doctors in A and C} = F_A + F_C = 240 + 300 = 540 \]

### Step 9: Final Ratio
The required ratio is:
\[ \text{Ratio} = \frac{690}{540} = \frac{23}{18} \]

### Conclusion
Thus, the final answer is:
\[ \text{The ratio of total male doctors in B and C to total female doctors in A and C is } \frac{23}{18}. \]