Question 19

Eight students wrote 3 papers.

To find how many students wrote 3 papers, we can set up an equation based on the total number of papers written. Let \( x \) represent the number of students who wrote 3 papers and \( y \) represent those who wrote 2 papers. We know that \( x + y = 20 \) and \( 3x + 2y = 48 \). Solving these equations reveals that 8 students wrote 3 papers.

A) 8

This option is correct. If 8 students wrote 3 papers, they contributed \( 3 \times 8 = 24 \) papers. The remaining \( 20 - 8 = 12 \) students would then have written \( 2 \) papers each, contributing \( 2 \times 12 = 24 \) papers. Thus, the total is \( 24 + 24 = 48 \), matching the given total.

B) 10

This option is incorrect. If 10 students wrote 3 papers, they would produce \( 3 \times 10 = 30 \) papers. The remaining \( 20 - 10 = 10 \) students writing 2 papers each would contribute \( 2 \times 10 = 20 \) papers, leading to a total of \( 30 + 20 = 50 \) papers, which exceeds the total of 48.

C) 12

This option is incorrect. If 12 students wrote 3 papers, the number of papers they would write is \( 3 \times 12 = 36 \). The remaining \( 20 - 12 = 8 \) students writing 2 papers would yield \( 2 \times 8 = 16 \) papers, resulting in a total of \( 36 + 16 = 52 \) papers, which is again greater than 48.

D) 14

This option is also incorrect. If 14 students wrote 3 papers, they would produce \( 3 \times 14 = 42 \) papers. The remaining \( 20 - 14 = 6 \) students writing 2 papers would contribute \( 2 \times 6 = 12 \) papers, leading to a total of \( 42 + 12 = 54 \) papers, which exceeds the total of 48.

Conclusion

The analysis confirms that Option A is the only correct choice, as it accurately satisfies both equations derived from the problem. All other options lead to totals that exceed the specified number of papers, demonstrating that they do not align with the conditions given in the problem.