#algebra, #current events - the supposed "meanest problem ever" on the SAT...

I read IFLScience on my Facebook feed daily and today there is a reference to the "meanest test problem ever" on the SAT. So without scrolling down, I decided to decided to tackle it and see what made it so hard. I gotta say I don't see the difficulty...

Question

In a class of p students, the average (arithmetic mean) of the test scores is 70.

In another class of n students, the average of the scores for the same test is 92.

When the scores of the two classes are combined, the average of the test scores is 86.

What is the value of p/n?

Answer

3/8

Analysis

Before we dive into the problem, let's first talk about means (also known as the average). We can find the mean using this general equation:

 

Let's see how this works. With the string of values 1, 2, 3, 4, 5, the sum of the values is 15 and the count is 5, so the mean is 3.



Another way to express this is to see that the sum of the values can be expressed as the product of the mean and the count:



and this is how we solve "the meanest problem ever".

We're told that for the class with p students:



and for the class with n students:



and when the class is put together:



Ok - so let's first work with the individual means and expand them:





and when we put the class together:



we can cross multiply:



and now we'll put p terms on one side and n terms on the other:







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Questions and comments always welcome!