(a) To determine the lowest mark Jenny can get in her fourth paper, we can use the formula for calculating the mean:
Mean = (Sum of all marks) / (Total number of papers)
Let's assume the mark she gets in her fourth paper is x. Given that the mean of the first three papers is 72% and she wants an overall mean of at least 75%, we can set up the following equation:
(72 + 72 + 72 + x) / 4 ≥ 75
Simplifying the equation:
216 + x ≥ 300
Subtracting 216 from both sides:
x ≥ 300 - 216
x ≥ 84
Therefore, the lowest mark Jenny can get in her fourth paper to achieve an overall mean of at least 75% is 84%.
(b) To calculate the highest possible mean she can get over all four papers, we need to maximize the marks on the first three papers and minimize the mark on the fourth paper.
Since the mean of the first three papers is given as 72%, to maximize the mean, Jenny needs to score 100% on each of these three papers. Therefore, the sum of the marks on the first three papers is 100 + 100 + 100 = 300.
Now, let's assume the mark she gets in her fourth paper is y. The equation for calculating the mean is:
Mean = (Sum of all marks) / (Total number of papers)
Therefore, the equation becomes:
(300 + y) / 4 = Mean
Given that the mean needs to be maximized, the denominator (total number of papers) needs to be minimized. Therefore, the highest possible mean occurs when there are only 3 papers.
So the highest possible mean she can get over all four papers is:
Mean = (300 + y) / 4 = (300 + y) / 3, where y is the mark on the fourth paper.
tl;dr
(a) The lowest mark Jenny can get in her fourth paper to achieve an overall mean of at least 75% is 84%.
(b) The highest possible mean she can get over all four papers is (300 + y) / 4 = (300 + y) / 3, where y is the mark on the fourth paper.