Answer: The teacher can grade between 12 and 24 English papers or between 18 and 36 math tests to meet the time constraint of spending over 3 hours but under 6 hours grading this weekend.
Step-by-step explanation:
Let's denote the number of English papers as E and the number of math tests as M. The time it takes to grade one English paper is 1/4 hour, and the time it takes to grade one math test is 1/6 hour.
The total time spent grading can be expressed as the sum of the time spent grading English papers and math tests:
Total time= E * 1/4 + M * 1/6
The teacher wants to spend over 3 hours but under 6 hours grading. So, the inequality representing the time constraint is:
3 < Total time<6
Now, let's find possible values for E and M that satisfy this inequality.
1. Finding the maximum time:
To maximize the time spent grading, we can consider the case where the teacher grades only English papers. In this case, M =0, and the total time is E times 1/4.
Max time: E * 1/4
Finding the minimum time:
To minimize the time spent grading, we can consider the case where the teacher grades only math tests. In this case, E =0, and the total time is M times 1/6.
Min time: M * 1/6
Now, we have the following inequalities:
3 <Max time<6
3 < E * 1/4 <6
12 < E <24
3 < Min time<6
3 < M * 1/6 <6
18 < M <36
So, the teacher can grade between 12 and 24 English papers or between 18 and 36 math tests to meet the time constraint of spending over 3 hours but under 6 hours grading this weekend.