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 difference between Total male doctors who represent C and total female doctors who represent B?

Options

A. 50
B. 70
C. 60
D. 80

Answer

To solve the problem step by step, we will break down the information given and use it to find the required values.

### Step 1: Define Variables
Let:
- \( x \) = Number of male doctors in Hospital A
- \( y \) = Number of male doctors in Hospital B

From the problem, we know:
- Total male doctors in A and B:
\[ x + y = 810 \]

### Step 2: Calculate Female Doctors in A and B
The number of female doctors in A and B is given as:
- Female doctors in A = \( \frac{2}{3}x \)
- Female doctors in B = \( \frac{2}{5}y \)

### Step 3: Average Female Doctors in A and B
The average number of female doctors in A and B is 210. Therefore:
\[ \frac{\frac{2}{3}x + \frac{2}{5}y}{2} = 210 \]
Multiplying both sides by 2:
\[ \frac{2}{3}x + \frac{2}{5}y = 420 \]

### Step 4: Find a Common Denominator
To solve the equation, we need a common denominator for the fractions. The LCM of 3 and 5 is 15. Thus, we can rewrite the equation:
\[ \frac{10}{15}x + \frac{6}{15}y = 420 \]
Multiplying through by 15 to eliminate the denominator:
\[ 10x + 6y = 6300 \]

### Step 5: Substitute for y
We can substitute \( y \) from the first equation \( y = 810 - x \) into the equation:
\[ 10x + 6(810 - x) = 6300 \]
Expanding this gives:
\[ 10x + 4860 - 6x = 6300 \]
Combining like terms:
\[ 4x + 4860 = 6300 \]
Subtracting 4860 from both sides:
\[ 4x = 1440 \]
Dividing by 4:
\[ x = 360 \]

### Step 6: Find y
Now substituting \( x \) back to find \( y \):
\[ y = 810 - 360 = 450 \]

### Step 7: Calculate Female Doctors in A and B
Now we can calculate the number of female doctors:
- Female doctors in A:
\[ \frac{2}{3}x = \frac{2}{3} \times 360 = 240 \]
- Female doctors in B:
\[ \frac{2}{5}y = \frac{2}{5} \times 450 = 180 \]

### Step 8: Calculate Female Doctors in C
Total female doctors who represent C are 25% more than total female doctors in A:
\[ \text{Female doctors in C} = 240 + 0.25 \times 240 = 240 + 60 = 300 \]

### Step 9: Calculate Male Doctors in C
Total male doctors who represent C are 33.33% more than female doctors in B:
\[ \text{Male doctors in C} = 180 + \frac{1}{3} \times 180 = 180 + 60 = 240 \]

### Step 10: Find the Difference
Finally, we need to find the difference between total male doctors who represent C and total female doctors who represent B:
\[ \text{Difference} = \text{Male doctors in C} - \text{Female doctors in B} = 240 - 180 = 60 \]

### Final Answer
The difference between total male doctors who represent C and total female doctors who represent B is **60**.

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