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.
Total doctors who represent A is what per cent more than total male doctor who represent B?

Options

A. `33 1/3%`
B. `30 1/3%`
C. `27 1/3%`
D. None of these

Answer

To solve the problem step by step, we will break down the information provided and calculate the required values.

### Step 1: Calculate Total Female Doctors in A and B
The average number of female doctors representing hospitals A and B is given as 210. Therefore, the total number of female doctors in A and B combined is:

\[ \text{Total Female Doctors in A and B} = 210 \times 2 = 420 \]

### Step 2: Determine the Number of Female Doctors in A and B
Let \( M_A \) be the number of male doctors in hospital A and \( M_B \) be the number of male doctors in hospital B. We know from the problem statement:

- Total male doctors in A and B: \( M_A + M_B = 810 \)
- Female doctors in A: \( F_A = \frac{2}{3} M_A \)
- Female doctors in B: \( F_B = \frac{2}{5} M_B \)

From the total female doctors calculated in Step 1, we can write:

\[ F_A + F_B = 420 \]

Substituting the expressions for \( F_A \) and \( F_B \):

\[ \frac{2}{3} M_A + \frac{2}{5} M_B = 420 \]

### Step 3: Solve the Equations
We have two equations:
1. \( M_A + M_B = 810 \)
2. \( \frac{2}{3} M_A + \frac{2}{5} M_B = 420 \)

To solve these equations, we can express \( M_B \) in terms of \( M_A \):

\[ M_B = 810 - M_A \]

Substituting this into the second equation:

\[ \frac{2}{3} M_A + \frac{2}{5} (810 - M_A) = 420 \]

Expanding and simplifying:

\[ \frac{2}{3} M_A + \frac{1620}{5} - \frac{2}{5} M_A = 420 \]

Finding a common denominator (15):

\[ \frac{10}{15} M_A + \frac{4860}{15} - \frac{6}{15} M_A = 420 \]

Combining like terms:

\[ \frac{4}{15} M_A + \frac{4860}{15} = 420 \]

Multiplying through by 15 to eliminate the fraction:

\[ 4 M_A + 4860 = 6300 \]

Solving for \( M_A \):

\[ 4 M_A = 6300 - 4860 \]
\[ 4 M_A = 1440 \]
\[ M_A = 360 \]

Now substituting back to find \( M_B \):

\[ M_B = 810 - 360 = 450 \]

### Step 4: Calculate Female Doctors in A and B
Now we can find the number of female doctors in A and B:

\[ F_A = \frac{2}{3} M_A = \frac{2}{3} \times 360 = 240 \]
\[ F_B = \frac{2}{5} M_B = \frac{2}{5} \times 450 = 180 \]

### Step 5: Calculate Total Doctors in A
Total doctors in hospital A:

\[ \text{Total Doctors in A} = M_A + F_A = 360 + 240 = 600 \]

### Step 6: Calculate Total Doctors in B
Total doctors in hospital B:

\[ \text{Total Doctors in B} = M_B + F_B = 450 + 180 = 630 \]

### Step 7: Find Total Doctors in A as a Percentage of Male Doctors in B
We need to find what percentage the total doctors in A is of the total male doctors in B:

\[ \text{Percentage} = \left( \frac{\text{Total Doctors in A} - \text{Total Male Doctors in B}}{\text{Total Male Doctors in B}} \right) \times 100 \]

Substituting the values:

\[ \text{Percentage} = \left( \frac{600 - 450}{450} \right) \times 100 = \left( \frac{150}{450} \right) \times 100 = \frac{1}{3} \times 100 = 33.33\% \]

### Final Answer
Total doctors who represent A is **33.33%** more than total male doctors who represent B.

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