(I) Two quadruple ratios  A:B:C:D and a:b:c:d are equal or in proportion or equivalent (choose your favorite term) and we write

A:B:C:D = a:b:c:d

when and only when there is a proportionality constant k such that k multiple of a term in one ratio gives the corresponding  terms in the other. Let say 

a =kA, b = kB,  c = kC and d = kD

The latter is equivalent to the simultaneous equalities 

a   A

=

b   B

=

c   C

=

d   D

which require all four fractions 

a   A

b   B

c   C

and

d   D

to have the same value - a value we have or may denote by k.

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The archaic double colon symbol in the expression A:B:C:D    ::  a:b:c:d  provides an alternative means to indicate A:B:C:D = a:b:c:d

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(II) Likewise, two triple ratios  A:B:C and a:b:c are equal or in proportion or equivalent (choose your favorite term) and we write

A:B:C = a:b:c

when and only when there is a proportionality constant k such that k multiple of a term in one ratio gives the corresponding  terms in the other. Let say 

a =kA, b = kB and c = kC.

The latter is equivalent to the simultaneous equalities 

a   A

=

b   B

=

c   C

which require all three fractions 

a   A

b   B

and

c   C

to have the same value - a value we have or may denote by k.

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The archaic double colon symbol in the expression A:B:C    ::  a:b:c  provides an alternative means to indicate A:B:C = a:b:c

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(III) Two double or ordinary ratios A:B and a:b are equal or in proportion or equivalent  we will write   A:B = a:b  when and only when   there is a constant k such that a =kA and b = kB. 

The latter is equivalent to the simultaneous equalities 

a   A

=

b   B

which require both fractions 

a   A

and

b   B

to have the same value - a value we have or may denote by k.

---

The archaic double colon symbol in the expression a:b  ::  c:d provides an alternative means to indicate a:b = c:d

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Note: Writing   A:B = a:b  when and only when   there is a constant k such that a =kA and b = kB implies

a   b

=

kA   kB

=

A   B

and hence 

a   b

=

A   B

The latter in turn implies the equalities 

a   A

=

b   B

which is equivalent to writing  A:B = a:b.  

Conclusion:  The ratios A:B and  a:b are equal when and only when the corresponding  fractions. 

a   b

=

A   B

are equal.

The conclusion provides an alternative way to start the description or characterization of ordinary or double ratios.  See the chapter Islands and Division of Knowledge common to site books  Pattern Based Reason  and  Three Skills for Algebra. 

Double ratios a:b of whole numbers are share many of the properties of fractions a/b. Double ratios a:b and c:d are equivalent when and only when the fractions a/b and c/d are equivalent or equal.  But double ratios like fractions cannot be added or combined (except in the case of equivalent ratios, to generate further equivalent ratios, a case beyond the scope of the present discussion).  Double a:b;  triple a:b:c and multiple ratios a:b:c: ... : z in general describe proportions and relative proportions. 

Remark:  While fractions a/b with whole number numerators a and whole number denominators b corresponds to the (double) ratio a:b, the fraction a/b is not the same as a ratio.