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%**.

This is very interesting. Would it be possible to build a simulation for this problem? the x% will be the variable. while the students could also change the other “variable” as sliders and I have a feeling it could be also represented graphically. What is so nice about GeoGebra that it provides such a strong visual connection between pictures, numbers, graphs, spreadsheets… Also the best thing is to teach students to DO the simulations because then THEY are doing the Math themselves.

Comment by Dani — 27 April 2009 @ 3:58 am

When I read this, you reminded me of a little problem I used to use with Physics students; indirectly related, the harmonic mean (although you can have a weighted harmonic mean). The simplest version; a person travels from A to B at 20 mi/h and immediately returns at 30 mi/h; what is the average speed for the total journey assuming that the time to turn around is negligible? You guessed it, just about every time I got the answer 25 mi/h! Thanks for the great posts.

Comment by The Math Maker — 29 April 2009 @ 2:31 pm