Аverage percentage

Average percentage

On Reddit, a person asked an interesting question about percentages:

Help with explaining negative compounding

Hey guys, not a maths teacher and I need to teach a concept related to work where something is negatively compounded.

For example:

Of a 1000 items 20% are rejected, leaving 800. At the next stage of those 800 30% are rejected, leaving 560. At next stage of the 560 all are rejected at 100% rejection.

The "average" rejection rate is NOT (20+30+100)/3 equalling 50%, it's 42% because "maths". I do actually have the formula and I personally understand why it's 42% instead of 50 but can anyone help me try to explain in a manner that makes it simple to understand WHY it's 42 and not 50?

I wrote my comment:

This is an incorrect application of mathematics. The average failure rate can only be calculated within one process. The average percentage between processes is absurd. Mathematically it looks like this:

1000*(1-0,42)=580
580*(1-0,42)=336,4
336,4*(1-0,42)=195,112

I showed that the proposed concept of "average percentage" does not allow you to get zero at the end of the calculation, which is not true. The man explained how he sees his decision:

I'm not sure I understand.

I understand the formula, as you have written, as it is what I use, and what I demonstrated above in a reply to an answer. But I don't understand what you mean about the average percentage between processes being absurd.

If I use my previous example:

First use = 1 (Used once, no rejection)
Second use = 1*(1-0.2) = 0.8 (20% rejected)
Third use = 0.8*(1-0.3) = 0.56 (30% rejected)
Fourth use 0.56*(1-1) = 0 (End of life product, 100% rejected)

Total average = 1/(1+0.8+0.56) = 0.42

In the end, using the average of 0.42, gives us an approximate product rejection, or if we want to call it something else, it's average "life", before it needs a brand new replacement.

Now I will show you what is wrong in this decision.

Let's go back to 1000 items. Absurdity is the summation of numbers with different units of measure. Percentages are taken from different numbers:

I use     100%=1000   1%=10
II use    100%=1000   1%=10
III use   100%=800    1%=8
IV use   100%=560    1%=5.6

To determine the average percentage, we are actually adding numbers with different units of measure. We received 42%, but it is not known from what number these percentages are to be calculated. It's like adding one dress to one gallon. We can get two. But what does the number two mean? Two gallons, two dresses, or two wet dresses?

1[gallon]+1[dress]=2[?]

To get wet dresses, you need to use multiplication:

1[gallon]*1[dress]=1[gallon*dress]

This math matches reality much better. By the way. In mathematics, the absence of a number is denoted by the number "zero". The absence of a letter in the grammar is indicated by a "space". Mathematicians say that zero is a number. Philologists have never said that a space is a letter. In units of measurement, it is more logical to use a space.

1[gallon*dress]=1[wet_dress]

Now let's look at percentages. To be correct, we must count the number of items rejected at each stage as a percentage of the total number of items.

I use      0 items      0%
II use    200 items   20%
III use   240 items   24%
IV use   560 items   56%

It's even better to count as a percentage the number of items that fail after each stage with a cumulative total. This will be a more realistic picture of what is happening.

I use      0 items      0%
II use    200 items   20%
III use   440 items   44%
IV use   1000 items  100%

Personally, I argue like this. In this case, it is not appropriate to calculate the average percentage.

The problem about the garden area

The problem about the garden area

Why such a strange image to task about the area of the garden? Because the problem itself, to put it mildly, very strange. Here's how it sounds:

The garden occupies 80 hectares. Apple trees occupy 5/8 of the area, and 31% of cherry trees. How many hectares of area under apple trees larger than the area under the cherry trees?

Let's first solve this problem for those students who want to write off the stupid decision, and only then talk about the strangeness of this problem.

The first action we define the area that is occupied by apple trees in the garden. To do this, the total area of the garden should be multiplied by the fractional expression of the area under apple trees.

80 * 5/8 = 50 ha

The second area of action is determined that the garden occupied cherry trees. Take the common garden area, multiplied by the number of cherry trees per cent and 100 per cent share. Interest on interest reduced and as a result we get an area of hectares.

80 * 31%: 100% = 24.8 ha

The area under apple trees we really get more than the area under the cherry trees. Takes away from smaller and larger area of the results.

50 - 24.8 = 25.2 ha

Answer: The area under apple trees on 25.2 hectares more than the area under the cherry trees.

Without checking any decision can be considered incorrect. How to check the solution to this problem? It is necessary to put together the area under apple trees and the area under the cherry trees. This result should be compared with the total area of the garden. If the sum is greater than the total area, so we decided not to challenge properly. If the sum is equal to or less than the total area, so our decision is correct.

50 + 24.8 = 74.8 hectares of less than 80 hectares

In most curious students immediately arises a natural question: what else is growing in this delicious daze, about what we were afraid to tell?

It was a children's nursery problem solution. Now the conversation for adults. This is the task of the textbook, which approved the Ministry of Education as an educational tool. The condition of this problem and at the same time used the percentage of fractional parts of a whole. Normal literate people'd never allow it. They used a ratio or percentage. Only an idiot is able to dump everything into one pile. The author of this task is illiterate idiot who either do not understand what he was doing, or for approval of a textbook command is ready to do anything. Stati, the quality of textbooks is very well characterizes the quality of all education. Our education is built on the principle of "one fools compose tutorials, others say they are fools".

For example, I'll write down the number on the same principle that is used by the author about the problem area of ​​the garden. I simultaneously use two forms of records: number and letter. That's what I got:

2 thousand three hundred 45

As you can see, only idiots can do so. Competent people will write down this number as follows:

2345 or two thousand three hundred forty-five

Why am I so vehemently opposed to such problems? Children - they are like a sponge that absorbs everything. If it is written in the book, then you can do so. As a result, we get the next portion of the idiots who write stupid textbooks that stupid leadership approves a flock of idiots. Just because they are so used to be taught.

What would you say there is not mathematics, but knowledge of mathematics lies not in the ability to accurately repeat all the things the teacher taught. Knowledge of mathematics - is the ability to competently and simply express their thoughts in the language of mathematics.