how a difference from the average value can give us the ratio?

[joshis1]

I have seen many ratio questions -- weighted average basically. For example - if one kind of rice having Rs 6 per kg is mixed with another kind of rice having Rs 15per kg to get an average price of Rs 10 per kg. Then what are the ratio?

I know that it is quite simple to solve - one easy way --

take the difference of average with the second categoery of rice -- i.e. 15 -10 i.e. 5
Now, take the difference of average with the first categoery of rice -- i.e. 10 -6 i.e. 4.
So, the ratio is 5:4.

I am trying to visualize this shortcut. Can somebody explain me , how come the difference gives us the value?
Please don't explain me in the equation terms. I want to visualize it conceptually. How the difference gives us the right value.

 

[Ishuda]

Let me start out in a equation terms from the word problem: You have an amount of the first category a priced at pa and an amount b of the second category priced at pb and, if you mix them together, the average price is P. In equation form this says
pa * a + pb * b = P (a+b)
or
(pa - P) a = (P- pb) b
or
\(\displaystyle \frac{pa-P}{P-pb} = \frac{b}{a}\)
which is your short cut.

Maybe this will work: If you think of this as a teeter-totter balancing act for example, if you are very heavy (much larger than the average) you will need to be much closer to the middle (have a smaller lever arm). If you are very light (much less than the average) you will need to be much further from the middle (have a larger lever arm). The distance you have to be from the middle is proportional to the difference in weights from the average.