In your class, you have scores of 73,85,76, and 92 on the first four of five tests. To get a grade of Upper C, the average of the first five tests scores must be greater than or equal to 70 and less than 80.
a) Solve an inequality to find the least score you can get on the last test and still earn a Upper C.
b) What score do you need if the fifth test counts as two tests?
a) To find the least score you can get on the last test and still earn an Upper C, we can use the inequality:
(73 + 85 + 76 + 92 + x) / 5 ≥ 70
Multiplying both sides of the inequality by 5, we have:
73 + 85 + 76 + 92 + x ≥ 70 * 5
326 + x ≥ 350
Subtracting 326 from both sides, we get:
x ≥ 24
Therefore, the least score you can get on the last test and still earn an Upper C is 24.
b) If the fifth test counts as two tests, we need to find the score required to maintain an average between 70 and 80 with seven total tests.
Let's represent the score on the fifth test as y. Now, we can calculate the required average using the inequality:
(73 + 85 + 76 + 92 + 2y) / 7 ≥ 70
Multiplying both sides by 7, we have:
73 + 85 + 76 + 92 + 2y ≥ 490
326 + 2y ≥ 490
Subtracting 326 from both sides, we get:
2y ≥ 164
Dividing both sides by 2, we have:
y ≥ 82
Therefore, if the fifth test counts as two tests, you would need a score of at least 82 on that test.
a) To find the least score you can get on the last test and still earn an Upper C, we need to find the average score of the first four tests and ensure that when combined with the score on the fifth test, it falls within the range of 70 to 80.
First, let's find the average of the first four tests:
(73 + 85 + 76 + 92) / 4 = 326 / 4 = 81.5
To calculate the least score you can get on the last test, we start by assuming it scores x, and then solve the inequality:
(73 + 85 + 76 + 92 + x) / 5 ≥ 70 and (73 + 85 + 76 + 92 + x) / 5 < 80
Multiplying both sides of the inequality by 5 gives:
73 + 85 + 76 + 92 + x ≥ 350 and 73 + 85 + 76 + 92 + x < 400
Combining like terms yields:
326 + x ≥ 350 and 326 + x < 400
Now, solving the inequalities:
x ≥ 350 - 326 and x < 400 - 326
x ≥ 24 and x < 74
Therefore, the least score you can get on the last test and still earn an Upper C is 24.
b) If the fifth test counts as two tests, we need to find the minimum overall average that falls within the range of 70 to 80, assuming the fifth test counts as two tests.
Let's calculate the combined score of the first four tests:
73 + 85 + 76 + 92 = 326
To calculate the required minimum average, we start by assuming the score on the fifth test counts as x and then solve the inequality:
(326 + 2x) / 6 ≥ 70 and (326 + 2x) / 6 < 80
Multiplying both sides of the inequality by 6 gives:
326 + 2x ≥ 420 and 326 + 2x < 480
Combining like terms yields:
2x ≥ 420 - 326 and 2x < 480 - 326
2x ≥ 94 and 2x < 154
Now, solving the inequalities:
x ≥ 94 / 2 and x < 154 / 2
x ≥ 47 and x < 77
Therefore, if the fifth test counts as two tests, you need a score between 47 and 76 on the fifth test to earn an Upper C.
To solve this problem, we'll follow these steps:
a) Solve an inequality to find the least score you can get on the last test and still earn an Upper C:
To find the least score needed on the last test to earn an Upper C, we'll set up an inequality and solve for the unknown score.
Let's assume the unknown score on the last test is x.
To find the average of all five scores, we add up all the scores and divide by the number of scores (which is 5 in this case).
Average of the five scores = (73 + 85 + 76 + 92 + x)/5
According to the given condition, the average must be greater than or equal to 70 and less than 80 for an Upper C grade:
70 ≤ (73 + 85 + 76 + 92 + x)/5 < 80
Now, we'll solve the inequality step by step:
70 ≤ (73 + 85 + 76 + 92 + x)/5 < 80
Multiply both sides of the inequality by 5 to get rid of the denominator:
350 ≤ 73 + 85 + 76 + 92 + x < 400
Combine like terms:
350 + 73 + 85 + 76 + 92 ≤ x + 73 + 85 + 76 + 92 < 400 + 73 + 85 + 76 + 92
Simplify:
676 ≤ x + 326 < 826
Subtract 326 from all sides of the inequality:
676 - 326 ≤ x + 326 - 326 < 826 - 326
Simplify:
350 ≤ x < 500
So, the least score you can get on the last test and still earn an Upper C is 350 or any value greater than 350 but less than 500.
b) What score do you need if the fifth test counts as two tests:
If the fifth test counts as two tests, we can consider it as two separate scores. Let's say the score for the fifth test is y.
Now, the total number of scores would be 6 (the first four tests, the fifth test as one score, and the fifth test as another score).
To find the average, we add up all the scores and divide by the number of scores:
Average of all six scores = (73 + 85 + 76 + 92 + y + y)/6
According to the given condition, the average must be greater than or equal to 70 and less than 80 for an Upper C grade:
70 ≤ (73 + 85 + 76 + 92 + y + y)/6 < 80
Now, we'll solve the inequality in a similar way as before:
70 ≤ (73 + 85 + 76 + 92 + 2y)/6 < 80
Multiply both sides of the inequality by 6 to eliminate the denominator:
420 ≤ 73 + 85 + 76 + 92 + 2y < 480
Combine like terms:
420 + 73 + 85 + 76 + 92 ≤ 2y + 73 + 85 + 76 + 92 < 480 + 73 + 85 + 76 + 92
Simplify:
746 ≤ 2y + 326 < 726
Subtract 326 from all sides of the inequality:
746 - 326 ≤ 2y + 326 - 326 < 726 - 326
Simplify:
420 ≤ 2y < 400
Divide all sides of the inequality by 2:
210 ≤ y < 200
So, the score you need on the fifth test (which counts as two tests) is 210 or any value greater than 210 but less than 200.
46
consider the total points. The total so far is 326. If the last grade is x, then you need
350 <= 326+x < 400
24 <= x < 74
If the 5th test counts as two, then it's like having six scores:
420 <= 326+2x < 480
94 <= 2x < 154
47 <= x < 77