Question 761707
A . . . .2 . . . . .B . . . 4 . . . . . C . . . 6
-- . = . --- . . . . -- . = . -- . . . . . -- .= .----
B . . . .3 . . . . .C . . . 5 . . . . . D . . . 7


The best way I think I can explain this is first to find the LCM of {{{3}}}, {{{5}}} and {{{7}}}, which, of course, is {{{3 * 5 * 7 = 105}}}.

Now express each of the fractions above with a denominator of {{{105}}}.

So, 
{{{A / B = 70 / 105}}}

{{{B / C = 84 /105}}}

{{{C / D = 90 / 105}}} . . . . . . . . . ( 1 )

In the last ratio, we have {{{C}}} as {{{90}}}. How can we express {{{B / C}}} as {{{x / 90}}}?

Well, 

{{{B / C = 84 / 105}}}, so if we want to express this as{{{ x / 90}}}, then

{{{84 / 105 = x / 90}}}

So {{{x = 84 (90 / 105) = 72}}}

Therefore we can express {{{B / C}}} as {{{72 / 90}}} . . . . . . . . ( 2 )

We have that {{{A / B = 70 / 105}}}, and now we want to get it so that {{{B}}} is {{{72}}}.

So {{{70 / 105 = y / 72}}}

Hence {{{y = 70 (72 / 105) = 48}}}

Therefore we can express {{{A / B}}} as {{{48 / 72}}} . . . . . . . . ( 3 )

Collecting the three ratios ( 1 ), ( 2 ) and ( 3 ) together, we have

A . . . 48 . . . . . B . . . .72 . . . . . C . . . 90
-- . = . ---- . . . . --- . = . ---- . . . . . -- .= .-----
B . . . 72 . . . . . C . . . .90 . . . . . D . . .105

From which you can see that {{{A : B : C : D = 48 : 72 : 90 : 105}}}

or, dividing by {{{3}}}, you get in lowest terms

{{{A : B : C : D = 16 : 24 : 30 : 35}}}