Cambridge101
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- Dec 16, 2021
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I am trying to see the link here. Can someone please explain!!
If you have two tests in a course you are doing, say you saore [imath]84~\&~76[/imath] out of [imath]100[/imath] what is your average on the two?I am trying to see the link here. Can someone please explain!!
Let's start with the definitions of the 2 terms. Please post them.I am trying to see the link here. Can someone please explain!!
This does not explain at all what I asked - I asked why!If you have two tests in a course you are doing, say you saore [imath]84~\&~76[/imath] out of [imath]100[/imath] what is your average on the two?
Well is it [imath]\dfrac{84+76}{2}~?[/imath] The mid-point is the average of the end-points.
[imath][/imath][imath][/imath]
Suppose that [imath]a<b\\\dfrac{a}{2}<\dfrac{b}{2}\\\dfrac{a}{2}+\dfrac{a}{2}<\dfrac{b}{2}+\dfrac{a}{2}\\a<\dfrac{a+b}{2}[/imath] likewise [imath]\dfrac{a+b}{2}<b[/imath] which means [imath]a<\dfrac{a+b}{2}<b[/imath]This does not explain at all what I asked - I asked why!
I think lev888 is right: In order to help you see the connection, we need to start with your understanding of what each concept means. For example, does "average", to you, just mean the formula (half the sum), or something more?I am trying to see the link here. Can someone please explain!!
Well,I think lev888 is right: In order to help you see the connection, we need to start with your understanding of what each concept means. For example, does "average", to you, just mean the formula (half the sum), or something more?
In particular, "why" is a tricky word; such a question can be answered in several very different ways. So we may need to make several attempts to explain, and your feedback will clarify what you think is lacking.
What does "why" or "the link" mean to you?
Yes this was a great way of looking at it - very, very much appreciated.I think you're pretty close to an understanding. I'll try saying it a couple other ways. (One thing I needed to see was whether you were specifically thinking of the formula for the midpoint of a line segment on a plane or in space (two or three dimensions), or were mostly thinking on a number line, as you say here. That will make it a little easier.
First, I'll talk about it symbolically. Rather than make one of the numbers by 0, let's take another example, say 3 and 15. The midpoint will be equidistant from these, so if it is x, then x-3 has to equal 15-x. We can solve that equation:
x - 3 = 15 - xx + x - 3 = 15 - x + x2x - 3 = 152x - 3 + 3 = 15 + 32x = 182x/2 = 18/2x = 9
As you look at that, you can see that we added 3 and 15, and divided by 2, which is how you average.
We can do the same thing with unknown numbers, a and b instead of 3 and 15:
x - a = b - xx + x - a = b - x + x2x - a = b2x - a + a = b + a2x = a + b2x/2 = (a + b)/2x = (a + b)/2
Or we can do the same sort of thing, in reverse, starting with the average. As you say, the average is the number you get when you share the total equally (which is why you add and divide by 2). Is this number equidistant from the two given numbers? To find out, we can evaluate the differences:
(a + b)/2 - a = a/2 + b/2 - a = (b/2 + (a/2 - a) = b/2 - a/2 = (b - a)/2b - (a + b)/2 = b - a/2 - b/2 = (b - b/2) - a/2 = b/2 - a/2 = (b - a)/2
So, the two distances are equal. And this reveals something useful: those distances are both half the difference between the given numbers.
And this, I think, is essentially what you came up with. An alternative way to find the average is to "split the difference". The average adds half the difference to the lower number, and subtracts half the difference from the higher number, so that it is the same distance from each.
Does that help at all?