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?

Rationale

There are 20 professors teaching.

Given that professors teach classes of 50 students and lecturers teach classes of 25 students, and knowing there are twice as many professors as lecturers, we can establish a system of equations to solve for the number of professors. The total population of professors, lecturers, and students sums to 1280, leading to the conclusion that there are 20 professors.

A (10) YOUR ANSWER

If there were 10 professors, then there would be 5 lecturers (since there are twice as many professors), leading to 10 * 50 + 5 * 25 = 500 + 125 = 625 students. Thus, the total would be 10 + 5 + 625 = 640, which does not match the given total of 1280.

B (15) YOUR ANSWER

With 15 professors, there would be 7.5 lecturers, which is not possible since the number of lecturers must be a whole number. This inconsistency rules out the possibility of having 15 professors.

C (20) CORRECT

Having 20 professors means there would be 10 lecturers. Calculating the total number of students gives us 20 * 50 + 10 * 25 = 1000 + 250 = 1250 students. Adding the number of professors and lecturers results in 20 + 10 + 1250 = 1280, confirming that this is the correct answer.

D (25) YOUR ANSWER

If there were 25 professors, then there would be 12.5 lecturers, which again is not feasible as the number of lecturers must be a whole number. Therefore, this option cannot be correct.

E (30) YOUR ANSWER

With 30 professors, there would be 15 lecturers, leading to 30 * 50 + 15 * 25 = 1500 + 375 = 1875 students. This total exceeds the given number of 1280, confirming that 30 professors is incorrect.

Conclusion

The problem involves a clear relationship between professors, lecturers, and the total student body. By setting up the equations based on the conditions given, we determined that having 20 professors fulfills all requirements and matches the total count of 1280 individuals, making it the only viable solution.

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?

There are 20 professors teaching.

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