Each of the 20 students in an English class wrote either 2 or 3 papers. If the students wrote a total of 48 papers, how many wrote 3 papers?
Eight students wrote 3 papers.
To solve this problem, we can set up a system of equations. Let \( x \) represent the number of students who wrote 2 papers and \( y \) the number who wrote 3 papers. The equations are \( x + y = 20 \) and \( 2x + 3y = 48 \). Solving these equations shows that \( y = 8 \).
This choice is correct because substituting \( y = 8 \) into the equations yields \( x + 8 = 20 \), resulting in \( x = 12 \). When we check the total papers written, \( 2(12) + 3(8) = 24 + 24 = 48 \), which matches the given total.
If 10 students wrote 3 papers, then \( y = 10 \). Substituting into \( x + y = 20 \) gives \( x = 10 \). The total papers would then be \( 2(10) + 3(10) = 20 + 30 = 50 \), which exceeds the total of 48 papers, making this choice incorrect.
Assuming 12 students wrote 3 papers results in \( y = 12 \). This leads to \( x + 12 = 20 \), giving \( x = 8 \). The total papers would be \( 2(8) + 3(12) = 16 + 36 = 52 \), again exceeding 48, thereby ruling out this option.
If 14 students wrote 3 papers, then \( y = 14 \). Thus, \( x + 14 = 20 \), which gives \( x = 6 \). The total papers would be \( 2(6) + 3(14) = 12 + 42 = 54 \), which is also greater than the total of 48 papers, making this choice invalid.
By analyzing the relationships between the number of students and the papers written, we find that only 8 students could have written 3 papers to achieve a total of 48 papers. The choices that exceed this total are invalid, confirming that the correct answer is indeed 8.
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