Ralph needs to earn a B in his Geology class. His current test scores are
82,83, 90, and 77. His final exam is worth
4 test scores. In order to earn a B​,
Ralph's average must lie between
80 and 89 inclusive. What range of scores can Ralph receive on the final exam to earn a
B in the​ course?
80 ≥ (4t + 82 + 83 + 90 + 77) / 8 ≥ 89
it works better if you use <=
oops...
89 ≥ (4t + 82 + 83 + 90 + 77) / 8 ≥ 80
To determine the range of scores Ralph can receive on the final exam to earn a B in the course, we need to calculate the minimum and maximum average scores that could be achieved with the given test scores.
First, let's find the minimum average score. Ralph's current test scores are 82, 83, 90, and 77. Since the final exam is worth 4 test scores, we can consider it as an additional test score.
To calculate the minimum average, we can assume the worst case scenario, which is Ralph scoring the lowest possible score on the final exam. In this case, the score would be 0.
Calculating the minimum average:
(82 + 83 + 90 + 77 + (0 * 4)) / 5 = 66.4
So, the minimum average Ralph can achieve is 66.4.
Next, let's find the maximum average score. Again, we consider the final exam as an additional test score and assume Ralph scores the highest possible score on the final exam. In this case, the score would be 100.
Calculating the maximum average:
(82 + 83 + 90 + 77 + (100 * 4)) / 5 = 94.4
So, the maximum average Ralph can achieve is 94.4.
Now, we know that Ralph's average must lie between 80 and 89 inclusive to earn a B in the course. Therefore, he needs his average score to be greater than or equal to 80 and less than or equal to 89.
To find the range of scores Ralph can receive on the final exam, we calculate the minimum and maximum possible scores that will give an average in the desired range.
Minimum possible score on the final exam:
Average = (82 + 83 + 90 + 77 + (x * 4)) / 5
80 ≤ ((82 + 83 + 90 + 77 + (x * 4)) / 5)
Multiply both sides by 5:
400 ≤ 82 + 83 + 90 + 77 + (x * 4)
400 - 82 - 83 - 90 - 77 ≤ x * 4
168 ≤ x * 4
Divide both sides by 4:
42 ≤ x
So, the minimum score Ralph can receive on the final exam is 42.
Maximum possible score on the final exam:
Average = (82 + 83 + 90 + 77 + (x * 4)) / 5
((82 + 83 + 90 + 77 + (x * 4)) / 5) ≤ 89
Multiply both sides by 5:
82 + 83 + 90 + 77 + (x * 4) ≤ 445
322 + (x * 4) ≤ 445
Subtract 322 from both sides:
(x * 4) ≤ 123
Divide both sides by 4:
x ≤ 30.75
Since the final exam score must be a whole number, the maximum score Ralph can receive on the final exam is 30.
Therefore, the range of scores Ralph can receive on the final exam to earn a B in the course is 42 to 30.