Study the following information carefully and answer the given questions.
The given data is regarding the number of doctors (D) and engineers (E) in Colleges A, B and C.
College A: Respective ratio between male doctors and male engineers is 12 : 5 and the difference between them, is 280. Total number of females (D+E) is 300. Number of male engineers is 30 more than the number of female engineers.
College B: Total number of males (D+E)=total number of females in college A (D + E). Respective ratio between total number of males (D+E) and total number of females (D +E) is 6:7 in College B. Respective ratio between the number of female doctors and the number of female engineers is 2:3. The number of male, doctors is one-third of the number of male engineers in College C.
College C : Number of male engineers is 1.5 times the number of female doctors. Total number of females (D +E) is 550. The number of female engineers is 70 more than the number of female doctors. Total number of males (D + E) is 760.
What is the average number of female doctors in colleges A, B and C?

Options

A. 160
B. 180
C. 200
D. 170

Answer

To find the average number of female doctors in colleges A, B, and C, we will follow these steps:

### Step 1: Determine the number of male doctors and male engineers in College A.
- The ratio of male doctors (D) to male engineers (E) in College A is 12:5.
- Let the number of male doctors be \(12x\) and the number of male engineers be \(5x\).
- The difference between the number of male doctors and male engineers is given as 280.

\[ 12x - 5x = 280 \implies 7x = 280 \implies x = 40 \]

- Therefore, the number of male doctors is \(12x = 480\) and the number of male engineers is \(5x = 200\).

### Step 2: Calculate the number of female doctors and engineers in College A.
- The total number of females (doctors + engineers) in College A is 300.
- The number of male engineers is 30 more than the number of female engineers. Let the number of female engineers be \(y\).

\[ 200 = y + 30 \implies y = 170 \quad (\text{female engineers}) \]

- Now, the number of female doctors can be calculated as:

\[ \text{Total females} = \text{Female doctors} + \text{Female engineers} \implies 300 = \text{Female doctors} + 170 \]

\[ \text{Female doctors} = 300 - 170 = 130 \]

### Step 3: Determine the number of female doctors in College B.
- The total number of males (doctors + engineers) in College B equals the total number of females in College A, which is 300.
- The ratio of males to females in College B is 6:7. Let the number of males be \(6a\) and females be \(7a\).

\[ 6a = 300 \implies a = 50 \]

- Thus, the number of females in College B is:

\[ 7a = 7 \times 50 = 350 \]

- The ratio of female doctors to female engineers is 2:3. Let the number of female doctors be \(2b\) and female engineers be \(3b\).

\[ 2b + 3b = 350 \implies 5b = 350 \implies b = 70 \]

- Therefore, the number of female doctors in College B is:

\[ 2b = 2 \times 70 = 140 \]

### Step 4: Determine the number of female doctors in College C.
- The total number of females (doctors + engineers) in College C is 550.
- The number of female engineers is 70 more than the number of female doctors. Let the number of female doctors be \(c\).

\[ \text{Female engineers} = c + 70 \]

\[ c + (c + 70) = 550 \implies 2c + 70 = 550 \implies 2c = 480 \implies c = 240 \]

- Thus, the number of female doctors in College C is 240.

### Step 5: Calculate the average number of female doctors in colleges A, B, and C.
- Now we have:
- Female doctors in College A = 130
- Female doctors in College B = 140
- Female doctors in College C = 240

- The average number of female doctors is calculated as:

\[ \text{Average} = \frac{\text{Total female doctors}}{\text{Number of colleges}} = \frac{130 + 140 + 240}{3} \]

\[ = \frac{510}{3} = 170 \]

### Conclusion
The average number of female doctors in colleges A, B, and C is **170**.