M07-07

Whenever an average or arithmetic mean problem comes up, and I have seen official questions in which the latter may not be defined, I find it useful to consider a basic way to derive an average:\(avg. = \frac{sum}{# items}\)
That may seem obvious, but standardized tests are good at presenting such a simple concept in disguise. I came across a question on a different test that was something like the following:

In a certain rating system, a user may assign a value of 1-100, inclusive, to something. After 10 ratings, the average score for Acme Corp. was 85.0. What is the lowest possible score the business can receive on the 11th rating and still end up with a 90 average after 20 ratings?

Seems simple enough, but that 11th rating in particular can look daunting. How can we separate the 11th rating from, say, the 16th, or the 19th? If you have read the problem, you might want to figure it out before reading any further. Done? All right, then let us continue our discussion with a few key insights.

1) The question is still based on nothing more than finding an average.
Key point: Make sure you know what the question is asking.

2) You have information on the first 10 ratings.
Key point: See what you can figure out based on those ratings. Here,\(85 = \frac{sum}{10}\)
A little quick handiwork, similar to the process you can use to solve the above problem, yields \(850 = sum\)

3) You can calculate the missing portion of the latter 10 ratings using the same basic average equation.
Key point: Run the average equation again to figure out the "target" total. \(90 = \frac{sum}{20}\)
Once again, algebraic manipulation yields \(90 * 20 = sum\), or \(1800 = sum\)

4) You can answer the question.
Key point: If 1800 points are needed to earn an average of 90 after 20 ratings, and after 10 ratings, there are only 850 points, then another 1800 - 850, or 950 points are needed. Thus, from the latter 10 ratings, if all of them were ratings of 100 except for the lowest, then that would produce 9 * 100, or 900 points, and the 11th rating (or whatever arbitrary number the question asked about) could be as low as 50 to make up the rest.

Good luck with your studies.

- Andrew

I think this is a high-quality question and I agree with explanation.

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.