A challenging AMC 10 problem involving some basic statistics

I was traveling for work the last two days and asked my son to work on some old AMC 10 problems rather than working in his geometry book. When I got back home tonight I asked him to pick one of the problems that gave him trouble for us to work through together. He picked #19 from the 2005 AMC 10 b:

The 2005 AMC 10 b

Here’s the problem:

On a certain math exam, 10\% of the students got 70 points, 25\% got 80 points, 20\% got 85 points, 15\% got 90 points, and the rest got 95 points. What is the difference between the mean and the median score on this exam?

This choice proved to be lucky since I’ve just started the section on basic statistics in Art of Problem Solving’s Prealgebra book with my younger son. It was a nice problem to work through with both of them.

We started by reading through the problem carefully and making sure that both boys understood what mean and median meant. The boys decided to solve the problem by assuming that 100 students took the exam. We solved for the median first:

Next we found the median by working through a long arithmetic calculation. Since the calculation itself isn’t really that interesting, I tried to focus more on building up number sense, and, in particular, on ways to make the calculation easier. My older son is definitely more comfortable working through calculations like this one, but I think my younger son was able to see a few good math ideas in action:

So, a nice problem giving some good practice in basic statistics as well as some good arithmetic review. It is also a nice illustration of why I like the problems from the old AMC 10 tests. This is #19 out of 25 problems on the AMC 10, meaning it is one of the more difficult problems. Not so difficult that the kids aren’t able to understand the solution, though. The AMC folks do a great job producing problems that strike that balance, which is something that I’d probably struggle mightily to do on my own.

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