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Quantities in proportion are also in proportion by composition

Given  a:b = c:d show (a+b)/b = (c+d)/d.


a:b = c:d can be written
a/b = c/d

Add 1 to both sides
a/b + 1 = c/d +1

So

(a+b)/b = (c+d)/d

Also by dividing both sides by (c+d) and multiplying both sides by b

(a+b)/(c+d) = b/d


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