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.
In another college D the number of females is 640 out of which `25%` are doctors and the number of male doctors in college is `12.5%` more than that in college C. What is the total number of doctors in college D?

Options

A. 630
B. 610
C. 570
D. None of those given as options

Answer

To solve the problem step by step, we will analyze the information provided for each college and derive the necessary values.

### Step 1: Analyze College A
1. **Male Doctors (MD) and Male Engineers (ME) Ratio**: Given as 12:5. Let MD = 12x and ME = 5x.
2. **Difference between Male Doctors and Male Engineers**:
\[ MD - ME = 280 \implies 12x - 5x = 280 \implies 7x = 280 \implies x = 40 \]
- Therefore, \(MD = 12x = 480\) and \(ME = 5x = 200\).
3. **Total number of females (D + E)**: Given as 300.
4. **Number of Female Engineers (FE) and Female Doctors (FD)**:
- Let \(FD + FE = 300\).
- Male Engineers are 30 more than Female Engineers: \(ME = FE + 30 \implies 200 = FE + 30 \implies FE = 170\).
- Therefore, \(FD = 300 - 170 = 130\).

### Step 2: Analyze College B
1. **Total number of males (D + E)**: Equals total number of females in College A, which is 300.
2. **Ratio of males to females in College B**: Given as 6:7.
- Let total males = 6y and total females = 7y.
- From the above, \(6y = 300 \implies y = 50\).
- Therefore, total males = 300 and total females = 350.
3. **Female Doctors (FD) and Female Engineers (FE) Ratio**: Given as 2:3.
- Let \(FD = 2z\) and \(FE = 3z\).
- Therefore, \(2z + 3z = 350 \implies 5z = 350 \implies z = 70\).
- Hence, \(FD = 140\) and \(FE = 210\).

### Step 3: Analyze College C
1. **Total number of females (D + E)**: Given as 550.
2. **Number of Female Engineers (FE) and Female Doctors (FD)**:
- Given that \(FE = FD + 70\).
- Let \(FD = a\), then \(FE = a + 70\).
- Therefore, \(a + (a + 70) = 550 \implies 2a + 70 = 550 \implies 2a = 480 \implies a = 240\).
- Hence, \(FD = 240\) and \(FE = 310\).
3. **Total number of males (D + E)**: Given as 760.
4. **Male Engineers (ME)**: Given as 1.5 times the number of Female Doctors:
\[ ME = 1.5 \times 240 = 360. \]
5. **Male Doctors (MD)**:
\[ MD + ME = 760 \implies MD + 360 = 760 \implies MD = 400. \]

### Step 4: Analyze College D
1. **Number of females**: Given as 640, and 25% are doctors.
\[ \text{Number of Female Doctors} = 25\% \text{ of } 640 = 0.25 \times 640 = 160. \]
2. **Number of Male Doctors in College D**: Given as 12.5% more than in College C.
\[ MD_{D} = MD_{C} + 12.5\% \text{ of } MD_{C} = 400 + 0.125 \times 400 = 400 + 50 = 450. \]

### Step 5: Total Number of Doctors in College D
1. **Total Doctors in College D**:
\[ \text{Total Doctors} = MD_{D} + FD_{D} = 450 + 160 = 610. \]

### Final Answer
The total number of doctors in College D is **610**.