inequalities

Question

Simon must maintain an average grade of 80% or better in his math class to be eligible to play sports. His math grades so far are: 80%, 84%, 76%, 77%. At this point, his grades make him ineligible to play. What is the lowest he score on his next grade to bring his average score up to an 80%.

Part A:
Write an inequality to represent the average of the five grades as greater than or equal to 80%.


Part B:
Use the inequality from Part A to prove what his lowest score could be on the 5th test.

Answer 1

x = his score for his next exam

 

(80 + 84 + 76 + 77 + x)/5 >= 80

(317 + x)/5 > = 80

317 + x >= 400

x >= 83

 

So he must score at least 83% to bring his average up to 80%.

 

Answer 2

An arithmetic average is (total of values) / (count of values).

 

So far, Simon has an average of:

     (80 + 84 + 76 + 77) / 4  =  79.25    [ineligible to play sports]

 

A.  In order to play sports, the average of 5 grades must be at least (that is, greater than or equal to) 80%:

    (80 + 84 + 76 + 77 + x) / 5  ≥ 80

 

B. 

     (80 + 84 + 76 + 77 + x) / 5  ≥ 80

     (80 + 84 + 76 + 77 + x) ≥ 400       [multiply both sides by +5 and retain sense of inequality]

       317 + x ≥ 400                            [collect terms]

        x ≥ 83            Simon must get a grade of 83 or greater on the next test.

 

 

Alternative method:

 

The difference of each grade from the required average will produce the same result.

   80  is    0 different from 80 

   84  is    +4   different from 80; 4 extra points

   76  is    -4    different from 80; 4 points lost

   77  is    -3    different from 80; 3 points lost

               ---------------------------------------

                 -3                           3 points short

  Simon must get 80+3 = 83 in order to make up the total of 3 points lost.