7. In a small college, professors teach classes of 50 students and lecturers teach classes of 25 students. There are twice as many professors as lecturers. The total number of professors, lecturers, and students is 1280. How many professors were teaching?
Answer: C
There are 20 professors teaching.
In the small college scenario, the problem reveals that the total number of professors is 20, which is determined by the constraints of class sizes and total population.
A) 10
Choosing 10 professors would imply 5 lecturers (since there are twice as many professors as lecturers), leading to 10 * 50 + 5 * 25 = 500 + 125 = 625 total students. This does not satisfy the total of 1280, making this option incorrect.
B) 15
If there were 15 professors, then there would be 7.5 lecturers, which is not possible since the number of lecturers must be a whole number. Therefore, this option is invalid as it does not meet the condition of having whole numbers of faculty.
C) 20
With 20 professors, there would be 10 lecturers. Calculating the total: 20 * 50 + 10 * 25 = 1000 + 250 = 1250 students. Adding the 30 faculty members gives a total of 1280, which aligns perfectly with the total stated in the problem, confirming this option as correct.
D) 25
If there were 25 professors, there would be 12.5 lecturers, which again cannot occur as the number of lecturers must be whole. Thus, this option is also invalid due to the non-integer count of lecturers.
E) 30
Selecting 30 professors would necessitate 15 lecturers. This would result in 30 * 50 + 15 * 25 = 1500 + 375 = 1875 students. This total exceeds the specified 1280, making this option incorrect.
Conclusion
The only viable solution that fits all the given criteria is that there are 20 professors teaching. All other options either lead to impossible fractional counts of lecturers or do not meet the total student count required by the problem. Thus, option C is definitively correct.