How to find proportion of Four numbers

If A : B = 2 : 3, B : C = 4 : 5 and      C : D = 6 : 7,    what is A  : D  ?

We shall find A : B : C : D

How to find A : B : C : D

Let  us mark these columns as follows 

                     1     2

A   :    B =     2    :   3 

B    :     C =    4    :   5

C     :     D =    6    : 7

The value of A can be calclulated by taking product  of all numbers  appearing under column (1).Therefore  A is product of  2 , 4 and 6

A = 2 x 4 x 6 = 48

The value of B can be calclulated by  product under column (1) beginning from second row and numbers  under column (2) for all previous rows. Therefore  B is product of  4 , 6 and 3

B = 4 x 6 x 3 = 72

Similarly C can be find out by taking  product of all numbers  under column (1) beginning from third row and terms under column (2) for all previous rows.Hence  C is product of  6 ,3 and 5

C = 6 x 3 x 5 = 90

D is result  of all terms appearing under column (2).

D = 3 x 5 x 7 = 105

=> A:B:C:D = 48 : 72 : 90 : 105

Ratio between any two or three variables from above.

A  : D = 48  :   105 = 16  :  35
This logic can be applied  to any number of rows

If A:B,  B:C,  C:D, D:E and E:F are  given, we can find A : B : C : D : E : F using above method.

Example :

A:B = 1:2

B:C = 2:7

C:D = 3:5

D:E = 3:8

E:F = 1:3

A = 1 × 2 × 3 × 3 ×1 = 18

B = 2 × 3 × 3 × 1 × 2 = 36

C =  3 × 3 × 1 × 2 × 7 =126

D = 3 × 1 × 2 × 7 × 5 = 210

E = 1 × 2 × 7 × 5 × 8 = 560

F = 2 × 7 × 5 × 8 × 3 = 1680

     A  :   B   :   C   :      D  :     E   :  F

= 18  :  36  : 126 : 210   : 560   : 1680
Now dividing  by 2 every terms,  we have 

= 9   : 18  : 63  : 105   :  280   :  840