During the quarter, the teacher gave students grades of 1, 2, 3, 4, 5. The average grade for a student was 3.5. a) What

Разъяснение:
a) To find the maximum fraction that "4"s could make up in a set of grades with an average of 3.5, we need to consider the worst-case scenario. In this case, the rest of the grades are all "5"s. Let"s assume we have N grades in total. The sum of all grades can be calculated as follows:

Sum = (N-1) * 3 + 4

We want the average to be 3.5, so we set up the equation:

( (N-1) * 3 + 4 ) / N = 3.5

Simplifying the equation, we get:

(N-1) * 3 + 4 = 3.5 * N

Expanding and simplifying further, we get:

3N - 3 + 4 = 3.5N

N - 4 = 0.5N

Multiplying both sides by 2, we get:

2N - 8 = N

N = 8

Therefore, the maximum number of "4"s that can make up the set of grades is 8/8, or 1 whole.

b) After replacing one "4" grade with one "3" and one "5," we need to find the highest possible average grade. By replacing one "4" grade with one "3" and one "5," we increase the sum of the grades by 3 - 4 + 5 = 4. To maximize the new average, we want to distribute this increase as evenly as possible among all the grades. Since the average is initially 3.5, there must be equal numbers of grades below and above the average. So, if we have N grades, half of them will be below 3.5 and half above 3.5.

The highest average can be achieved when N is even. Let"s assume N = 8. Initially, the sum of the grades is (8 * 3.5) = 28. By replacing one "4" with one "3" and one "5," the new sum becomes 28 + (3 - 4) + 5 = 32. Therefore, the highest possible average is 32/8 = 4.

c) Similarly, to find the highest possible average grade after replacing each "4" grade with one "3" and one "5," we again want to distribute the increase evenly among all the grades. Assume N is even. Initially, the sum of the grades is N * 3.5, and after replacing each "4" with one "3" and one "5," the new sum becomes N * 3 + (3 - 4) * N/2 + 5 * N/2 = N * 3.5 + N/2 = 4N. Therefore, the highest possible average after this replacement is 4.

Демонстрация:
a) The maximum fraction of "4"s in the set of grades is 1/8.
b) The highest possible value for the average grade after replacing one "4" grade is 4.
c) The highest possible value for the average grade after replacing each "4" grade is 4.

Совет: To solve problems like these, it is important to carefully analyze the given information and think about the possible scenarios. Construct equations based on the given conditions and solve them step by step. Also, always check if your answers make logical sense in the context of the problem.

Задание для закрепления: In a set of 20 grades, the average grade is 4.5. The teacher wants to replace each "5" grade with two grades: one "4" and one "5". Find the highest possible value for the average grade of the students after this replacement.