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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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