The final grade for a particular course is calculated by dividing total points earned by total points available. There are 120 points available in the first half of the course, and 140 points in the second half of the course. If a student earns 81% in the first half of the course, how many points must the student earn in the second half in order to earn an 80% B final grade?

120 * 0.81 = 97.2

260 * 0.8 = 208

208 - 97 = 111

There are 2 halves in the course:

1st half has 120 points
2nd half has 140 points
-- for a grand total of 260 points

If a student earns 81% of the points in the 1st half, she earns 0.81 * 120 points
-- 0.81 * 120 = 97.2 points

If there are 260 total points in the class, and the student wants to earn a B final grade, that is 0.8 * 260 points
-- 0.8 * 260 = 208 points

208 - 97.2 = 110.8 points
-- the student still needs 110.8 or 111 points from the second half

To find out how many points the student must earn in the second half of the course in order to earn an 80% final grade, we need to first determine the total points that the student needs to earn overall for the course.

Let's assume the student needs to earn x points in the second half of the course.

Next, we can calculate the total points available for the course by summing up the points available in the first and second half:

Total points available = points available in the first half + points available in the second half
Total points available = 120 points + 140 points
Total points available = 260 points

Since we know that the final grade is calculated by dividing total points earned by total points available, we can set up the following equation:

Final grade = (points earned / total points available) * 100

In this case, the student needs to earn an 80% B final grade, so we can write:

80 = (points earned / 260) * 100

We can solve this equation for points earned:

80 * 260 = points earned * 100
20800 = points earned * 100

To isolate points earned, we divide both sides of the equation by 100:

20800 / 100 = points earned
208 = points earned

Now we know that the student needs to earn a total of 208 points in the course to achieve an 80% final grade.

Since the student has already earned 81% in the first half of the course, we can calculate the points earned in the first half as a fraction of points available:

Points earned in the first half = (81/100) * 120
Points earned in the first half = 97.2 points

To find out how many points the student must earn in the second half, we subtract the points earned in the first half from the desired total points:

Points needed in the second half = Total points needed - Points earned in the first half
Points needed in the second half = 208 - 97.2
Points needed in the second half = 110.8 points

Therefore, the student must earn 110.8 points in the second half of the course in order to earn an 80% B final grade.