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#1 2010-12-05 23:32:55

Mariner
Member
Registered: 2010-06-16
Posts: 35

Averages in your head

Hi, I was wondering if someone can give tell me some tops or tricks to calculating averages in ones head.
For eg

Average of 61 and 18 = (61+18)/2 = 79/2 = 39.5

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#2 2010-12-06 00:51:50

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Averages in your head

It doesn't really apply to the example you gave, but you can sometimes "scale" the numbers you want to average, find the average of the scaled numbers, and then reverse the scale to get the right answer.

eg.

Find the average of 55, 54 and 59.

You can scale each of these numbers by taking 50 away. The average of 5, 4 and 9 is 18/3=6, so the answer is 6+50 = 56.

You can also scale by multiplying and dividing.

To find the average of 16, 24 and 36, it could be easier to find the average of 4, 6 and 9, then multiply the answer by 4.


Why did the vector cross the road?
It wanted to be normal.

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#3 2010-12-06 02:21:09

bob bundy
Administrator
Registered: 2010-06-20
Posts: 8,168

Re: Averages in your head

hi Mariner,

I agree with all that mathsyperson has said.

Have a look at my graph below.  It illustrates that (5 + 4 + 9) / 3 = 6.

The high column has been 'chopped off' at 6, and the blocks re-distributed on 5 and 4 so that all columns are now 6.

This is what 'finding an average' does.  It shares out the amounts equally. 

If you keep that picture in your head, you can see why adding , say, 50 to every column takes the average up by 50 as well.

And if every column is doubled in height the average is doubled. 

[Just imagine the 'y' axis is doubled so that the graph shows (10 + 8 + 18) / 3 = 12 instead.]

This will always be true for the mean average so you can use the picture to make up your own short cuts.

eg Here's one I just thought up:

76.3,   77.4,   76.1

Take 0.8 off of 77.4 = 76.6

Give this out to the other numbers like this ..... 76.3 + 0.3 = 76.6 ................  76.1 + 0.5 = 76.6

Now I've re-distributed bits of the highest number so that all three are now the same.

Average = 76.6

Bob

View Image: averages.JPG

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#4 2010-12-06 06:55:22

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Averages in your head

Hi Mariner;

I think it is useful to learn the techniques of quick addition and division. Sort of like Vedic math. There are other systems as well. With them you will find you can add a long column of 3 digit numbers, multiply 3 digit numbers by 2 digit numbers, square 3 digit numbers, all in your head. When in my 20's I could do that naturally.  Google for them and you will find what you need.

An important point, one's ability for that decreases with age, like blindfold or blitz chess.

For big lists, use a calculator.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#5 2010-12-07 16:30:02

Mariner
Member
Registered: 2010-06-16
Posts: 35

Re: Averages in your head

Thanks for the help. "It doesn't really apply to the example you gave", i'm sorry but I disagree with you. I'm using aptitude software which wants me to find the average by the method stated in the example in my head. I also can't round up or down. I'll try the what has been said. Thanks!

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#6 2010-12-07 17:17:28

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Averages in your head

Hi Mariner;

If you have to do it exactly by the method you gave for your problem and you cannot use any of the ideas given here. Then you must practice until you get quick and accurate. The average human should be able to add up 2,3, or 4, 2 digit numbers in his head and take the average after a little practice.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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