questioner wrote:

I don't really understand the reasoning behind the shorter method.

First of all, I'd like to note, that this shortcut is quite advanced one. It can be used here only if you see it as a glimpse and are very familiar with it. So if you don't get it right away - it's better and faster to use the general method, which is provided in the explanation.

1. We have some quantity of D-marks (denote by

*x*), each corresponds to "1 point". We have about the same quantity of A-marks (

*x* as well), each corresponds to "4 points". Their average will be (

*x* × 1 +

*x* × 4)/(2

*x*) = (1 + 4)/2 for ANY

*x*.

**Here is the tricky moment #1**. You need to feel this "about the same quantity". What can the difference be: 2, 10, 50? It depends on the quantities.

Let's say 98 of 1s and 100 of 4s are about the same, but 4 of 1s and 2 of 4s are NOT.

98 of 0.00001s and 100 of 2s are NOT good as well.

2. We did the same for C-marks and B-marks and then we've combined the results, because they were "about the same".

**Here is the tricky moment #2**. We've skipped all the rest of the calculations, because the points were "about the same" in the both case.

If they were different, let's say 2 and 3, this shortcut would NOT work. One group could have contained 100 marks (3 average), while the other one could have contained 10 marks (2 average). The combined average is NOT (2 + 3)/2 = 2.5, but it IS (3 × 100 + 2 × 10)/110 ≈ 2,9.

ADVICE:

If you don't see a shortcut, or not sure if it can be applied, then use the general method - it will deliver the desired results.

If you're low on time and see a shortcut, but not so sure if it can be applied - your decision depends on how much time you have left and how sure you are about your shortcut.

USEFUL FACT:

A relevant fact about average values, that can lead to a shortcut.

An average value always lies between the maximum and minimum values. E.g. we have some quantity of 2s and 4s - the average value will be between 2 and 4. If there're more 4s - it will be closer to 4. If there're more 2s - it will be closer to 2.