I loved this question. You can ask it in almost any class.

If Johnny has a 82% average for the 1st 9weeks and a 75% average for the 2nd 9weeks, what grade would he have to get on the final to receive 80% semester grade in this class. The grades are weighted as follows: each 9 weeks average counts 40% and the final test counts 20%.

Source: MathForum@Drexel

How did I find it? I was working on a Build your Own Simulator Kit using the freeware GeoGebra. To build the simulator, the student must make the ball go from point A to point B as * s* goes from 0 to 1. How to explain that this is: P=A*(1-

*)+B**

**s***? You can see here. Then a colleague of mine said “This is*

**s***weighted averages*and can be used in algebra, probability and geometry and can be expanded to n weights. Too bad we don’t talk about that anymore. Those problems are really rich.” So I looked around to see what he was talking about and found this problem. There is so much logic and usefulness in this problem – but not really any kind of special mathematics. Too bad we label it with such an awful name: weighted averages. It is just a good question.

Solution:

1. Check that the grading system makes sense: 2 × 9 week + 1 × final test = 2 × 40% +1 × 20% = 100%. Right.

2. Let * x*=

*grade on final test*.

Each 9 week grade counts 40% and final test 20% so we have: 82% × 40% + 79% ×40% +

*×20%*

**x**What does this expression equal? Well, we want an 80% total average at the end.

But we cannot write

*Expression*=80% because it is obvious that we need another “%”.

(This is actually the hard part.) We have not used “at the end“. What does “at the end” mean? It means 100%.

So our equation is: 82% × 40% + 79% ×40% + * x* × 20% = 80% × 100%.

Cancel all percentages and solve to get ** x=79%**.