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 total number of engineers (male ± female) in colleges A, B and C together?

Options

A. 1390
B. 1300
C. 1510
D. 1430

Answer

To solve the problem step-by-step, we will analyze the information provided for each college and calculate the required values.

### Step 1: Analyze College A
1. **Given Ratios and Differences**:
- Ratio of male doctors (D) to male engineers (E) = 12:5
- Difference between male doctors and male engineers = 280

2. **Let the number of male doctors be 12x and male engineers be 5x**:
\[ 12x - 5x = 280 \implies 7x = 280 \implies x = 40 \]

3. **Calculate the number of male doctors and engineers**:
- Male doctors = \(12x = 12 \times 40 = 480\)
- Male engineers = \(5x = 5 \times 40 = 200\)

4. **Total number of females (D + E) in College A**:
- Total females = 300
- Male engineers = 200, so:
\[ \text{Female engineers} = \text{Male engineers} - 30 = 200 - 30 = 170 \]
- Female doctors = Total females - Female engineers = \(300 - 170 = 130\)

### Step 2: Analyze College B
1. **Total number of males (D + E) in College B**:
- Total males = Total females in College A = 300

2. **Ratios of males to females**:
- Ratio of males to females = 6:7
- Let males = 6y and females = 7y:
\[ 6y + 7y = 300 \implies 13y = 300 \implies y = \frac{300}{13} \approx 23.08 \]

3. **Calculate total males and females**:
- Males = \(6y = 6 \times 23.08 \approx 138.46\) (round to 138)
- Females = \(7y = 7 \times 23.08 \approx 161.54\) (round to 162)

4. **Female doctors and engineers ratio**:
- Ratio = 2:3
- Let female doctors = 2z and female engineers = 3z:
\[ 2z + 3z = 162 \implies 5z = 162 \implies z = \frac{162}{5} = 32.4 \]
- Female doctors = \(2z = 64.8\) (round to 65)
- Female engineers = \(3z = 97.2\) (round to 97)

### Step 3: Analyze College C
1. **Total number of females (D + E)**:
- Total females = 550
- Female engineers = Female doctors + 70
- Let female doctors = a, then female engineers = a + 70:
\[ a + (a + 70) = 550 \implies 2a + 70 = 550 \implies 2a = 480 \implies a = 240 \]
- Female doctors = 240, Female engineers = 310

2. **Total males (D + E)**:
- Total males = 760

3. **Male engineers**:
- Male engineers = 1.5 times female doctors = \(1.5 \times 240 = 360\)

4. **Calculate male doctors**:
- Male doctors = Total males - Male engineers = \(760 - 360 = 400\)

### Step 4: Calculate Total Engineers
1. **Total engineers in College A**:
- Male engineers = 200, Female engineers = 170
- Total engineers in A = \(200 + 170 = 370\)

2. **Total engineers in College B**:
- Male engineers = 138, Female engineers = 97
- Total engineers in B = \(138 + 97 = 235\)

3. **Total engineers in College C**:
- Male engineers = 360, Female engineers = 310
- Total engineers in C = \(360 + 310 = 670\)

4. **Total engineers in all colleges**:
\[ \text{Total engineers} = 370 + 235 + 670 = 1275 \]

### Final Answer
The total number of engineers (male + female) in colleges A, B, and C together is **1275**.