Solve application problems using linear equations.

Question

Suppose that you are taking a course that has 4 tests. The first three tests are for 100 points each and the fourth test is for 200 points. To get a B in the course, you must have an average of at least 80 percent on the 4 tests. Your scores on the first 3 tests were 83, 71, and 74. What is the minimum score you need on the fourth test to get a B for the course?

Answer 1

There are two possible cases here: (1) equally weighted grades or (2) proportionally weighted grades

Case 1: Equally weighted grades

If we assume that all 4 of the tests are equally weighted then,the following inequality needs to be true in order to receive 80 percent (0.80) or greater as an average, where x is the grade on test # 4, out of 200pts.

1. (83/100 + 71/100 +74/100 + x/200) / 4 >= 0.80 .... average of 4 equally weighted grades should be at least 80 percent
2. (83/100 + 71/100 +74/100 + x/200) >= 3.20 .... multiplying both sides by 4
3. 83 + 71 + 74 + x/2 >= 320 .... multiplying both sides by 100
4. x/2 >= 320 - 83 - 71 - 74 ..... combining all constant expressions on the right hand side
5. x/2 >= 92 .... multiplying both sides by 2
6. x > =184 ... so at least 184 out of 200 pts is needed to get a B grade

Case 2: Proportionally weighted grades

On the other hand, if each test is weighted according to the number of points it is out of, then the following inequality has to be solved.

1. (83 + 71 + 74 + x)/(100 +100 + 100 +200) >= 0.80 ... Total points earn divided by the total points offered should be at least 80 percent
2. (228 + x)/500 >= 0.80 ... combining all constant terms in numerator and denominator
3. (228 + x) >= 400 ... multiplying both sides by 400
4. x >= 172 .... so at least 172 out of 200 pts is needed to get a B grade

Answer 2

The average would be [(score1)(100) + (score2)(100) + (score3)(100) + (score4)(200)]/(100 + 100 + 100 + 200)

So, if x = the score on test 4:

80 = (83*100 + 71*100 + 74*100 + 200x)/500

80*500 = 8300 + 7100 + 7400 + 200x

40000 = 22800 + 200x

17200 = 200x

x = 17200/200 = 86