All I did was write "A B C D" on my noteboard with large spaces.

Start with B and C. Let's give them easy number. Let B = 2, C = 6
that would make D=4 (2 plus 6, divided by 2)
that would make A=5

(C-A)/(A-B)
= (6-5)/(5-2)
=1/3
which can not be further reduced so that's my answer.

Just for kicks, lets try something else.

Let's make A=100
So C and D have to average 100
Let's make C=70 and D=130. Remember, since we are averaging only 2 numbers, we just have to make it so that if one number is greater than the average, the other number needs to be less than the average by the same amount.
So now B and C need to average to 130
This forces B to = 190

(C-A)/(A-B)
= (70-100)/(100-190)
=(-30)/(-90)
both top and bottom or negative, so its a positive (or if you will, multiply top and bottom by -1
=30/90
reduce by dividing top and bottom by 30
=1/3

For an algebraic solution, you can do the following:

we know

(C+B)/2 = D
(D+C)/2 = A

we are asked for

(C-A)/(A-B)

we need to relate A, B and C while removing D, we can substitute the first known equation into the second to get

(((C+B)/2)+C)/2 = A
(3C+B)/4 = A

subbing that for A above we are left with C and B only

= (C - (3C+B)/4 ) / ((3C+B)/4 - B)
= (4C - 3C - B) / (3C + B - 4B)
= (C-B) / (3C - 3B)
= 1/3

I don't trust myself with VICs, I can never pick the right numbers so algebra is easier for me. But if this confused you then go with the post before it's much simpler to follow I'm sure and shorter.

i agree, in some cases algebraic solutions are better than the ones in which you have to pick the numbers.

Coolgirl26: Just draw it on a number line and it becomes clear..
check the picture below... you'll understand what I mean..




You can see as per equations given... that the picture depicts it correctly..
and so the answer is pretty simple and comes out of the drawing itself..

Given
1)B < A

2) D is average of C and B So D lies between C and B

3) A is average of D and C so A lies between C and D

with above three statements, the letters are arranged as B < D < A < C

Now plotting on number system :-

B----D--A--C

as can be seen C-A / A - B = 1/3

Given:

C + B = 2D -- (1)
D + C = 2A -- (2)

since we have relationship b/w A,B,and C, so eliminate D.

multiply eq2 by 2 and add eq1 and eq2

C + B + 2C= 4A

3C - 3A = A - B

(C - A)/(A - B)= 1/3