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Question

What quantity must be added to each term of the ratio x : y so that it may become equal to c : d ?

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Solution

Let the quantity be k.
Then,

fraction numerator a plus k over denominator b plus k end fraction equals c over d
⇒ (a + k)d = (b + k)c
⇒ ad + dk = bc + ck
⇒ ad - bc = ck - dk
⇒ ad - bc = k(c - d)
∴ k = fraction numerator a d minus b c over denominator c minus d end fraction equals k

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